Properties

Label 32-322e16-1.1-c1e16-0-0
Degree $32$
Conductor $1.336\times 10^{40}$
Sign $1$
Analytic cond. $3.64861\times 10^{6}$
Root an. cond. $1.60349$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 6·3-s + 28·4-s − 48·6-s − 48·8-s + 11·9-s + 168·12-s + 6·16-s − 88·18-s − 4·23-s − 288·24-s + 21·25-s − 6·27-s + 16·29-s − 24·31-s + 168·32-s + 308·36-s + 32·46-s + 30·47-s + 36·48-s − 29·49-s − 168·50-s + 48·54-s − 128·58-s + 36·59-s + 192·62-s − 384·64-s + ⋯
L(s)  = 1  − 5.65·2-s + 3.46·3-s + 14·4-s − 19.5·6-s − 16.9·8-s + 11/3·9-s + 48.4·12-s + 3/2·16-s − 20.7·18-s − 0.834·23-s − 58.7·24-s + 21/5·25-s − 1.15·27-s + 2.97·29-s − 4.31·31-s + 29.6·32-s + 51.3·36-s + 4.71·46-s + 4.37·47-s + 5.19·48-s − 4.14·49-s − 23.7·50-s + 6.53·54-s − 16.8·58-s + 4.68·59-s + 24.3·62-s − 48·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{16} \cdot 7^{16} \cdot 23^{16}\)
Sign: $1$
Analytic conductor: \(3.64861\times 10^{6}\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{16} \cdot 7^{16} \cdot 23^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.001788154811\)
\(L(\frac12)\) \(\approx\) \(0.001788154811\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} )^{8} \)
7 \( 1 + 29 T^{2} + 480 T^{4} + 5389 T^{6} + 43787 T^{8} + 5389 p^{2} T^{10} + 480 p^{4} T^{12} + 29 p^{6} T^{14} + p^{8} T^{16} \)
23 \( 1 + 4 T + 12 T^{2} + 24 T^{3} + 122 T^{4} + 612 T^{5} + 4976 T^{6} + 58228 T^{7} + 108771 T^{8} + 58228 p T^{9} + 4976 p^{2} T^{10} + 612 p^{3} T^{11} + 122 p^{4} T^{12} + 24 p^{5} T^{13} + 12 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
good3 \( ( 1 - p T + 8 T^{2} - 5 p T^{3} + 26 T^{4} - 4 p^{2} T^{5} + 49 T^{6} - 10 p^{2} T^{7} + 145 T^{8} - 10 p^{3} T^{9} + 49 p^{2} T^{10} - 4 p^{5} T^{11} + 26 p^{4} T^{12} - 5 p^{6} T^{13} + 8 p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} )^{2} \)
5 \( 1 - 21 T^{2} + 8 p^{2} T^{4} - 1281 T^{6} + 7492 T^{8} - 44982 T^{10} + 254441 T^{12} - 263172 p T^{14} + 6575701 T^{16} - 263172 p^{3} T^{18} + 254441 p^{4} T^{20} - 44982 p^{6} T^{22} + 7492 p^{8} T^{24} - 1281 p^{10} T^{26} + 8 p^{14} T^{28} - 21 p^{14} T^{30} + p^{16} T^{32} \)
11 \( 1 + 4 p T^{2} + 868 T^{4} + 10894 T^{6} + 117599 T^{8} + 1319789 T^{10} + 16514609 T^{12} + 231942793 T^{14} + 2908533967 T^{16} + 231942793 p^{2} T^{18} + 16514609 p^{4} T^{20} + 1319789 p^{6} T^{22} + 117599 p^{8} T^{24} + 10894 p^{10} T^{26} + 868 p^{12} T^{28} + 4 p^{15} T^{30} + p^{16} T^{32} \)
13 \( ( 1 - 37 T^{2} + 1044 T^{4} - 18335 T^{6} + 280823 T^{8} - 18335 p^{2} T^{10} + 1044 p^{4} T^{12} - 37 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
17 \( 1 - 35 T^{2} + 219 T^{4} + 11480 T^{6} - 220090 T^{8} - 1178205 T^{10} + 59357296 T^{12} + 99415540 T^{14} - 17599467771 T^{16} + 99415540 p^{2} T^{18} + 59357296 p^{4} T^{20} - 1178205 p^{6} T^{22} - 220090 p^{8} T^{24} + 11480 p^{10} T^{26} + 219 p^{12} T^{28} - 35 p^{14} T^{30} + p^{16} T^{32} \)
19 \( 1 - 7 p T^{2} + 9636 T^{4} - 492149 T^{6} + 19605224 T^{8} - 640268790 T^{10} + 17635326721 T^{12} - 416319330952 T^{14} + 8496357233013 T^{16} - 416319330952 p^{2} T^{18} + 17635326721 p^{4} T^{20} - 640268790 p^{6} T^{22} + 19605224 p^{8} T^{24} - 492149 p^{10} T^{26} + 9636 p^{12} T^{28} - 7 p^{15} T^{30} + p^{16} T^{32} \)
29 \( ( 1 - 4 T + p T^{2} + 68 T^{3} - 335 T^{4} + 68 p T^{5} + p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
31 \( ( 1 + 12 T + 120 T^{2} + 864 T^{3} + 5289 T^{4} + 34809 T^{5} + 233559 T^{6} + 1606263 T^{7} + 9513677 T^{8} + 1606263 p T^{9} + 233559 p^{2} T^{10} + 34809 p^{3} T^{11} + 5289 p^{4} T^{12} + 864 p^{5} T^{13} + 120 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
37 \( 1 + 22 T^{2} - 1442 T^{4} - 144646 T^{6} + 126101 T^{8} + 240960367 T^{10} + 10207480289 T^{12} - 214540022509 T^{14} - 17622239692205 T^{16} - 214540022509 p^{2} T^{18} + 10207480289 p^{4} T^{20} + 240960367 p^{6} T^{22} + 126101 p^{8} T^{24} - 144646 p^{10} T^{26} - 1442 p^{12} T^{28} + 22 p^{14} T^{30} + p^{16} T^{32} \)
41 \( ( 1 - 261 T^{2} + 32124 T^{4} - 2407575 T^{6} + 119797031 T^{8} - 2407575 p^{2} T^{10} + 32124 p^{4} T^{12} - 261 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 - 84 T^{2} + 960 T^{4} + 107073 T^{6} - 6121135 T^{8} + 107073 p^{2} T^{10} + 960 p^{4} T^{12} - 84 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
47 \( ( 1 - 15 T + 258 T^{2} - 2745 T^{3} + 30366 T^{4} - 266016 T^{5} + 2368407 T^{6} - 17617218 T^{7} + 131996741 T^{8} - 17617218 p T^{9} + 2368407 p^{2} T^{10} - 266016 p^{3} T^{11} + 30366 p^{4} T^{12} - 2745 p^{5} T^{13} + 258 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
53 \( 1 + 113 T^{2} + 5770 T^{4} + 339547 T^{6} + 17313788 T^{8} + 818611682 T^{10} + 46552058051 T^{12} + 1264344130294 T^{14} + 12284303763469 T^{16} + 1264344130294 p^{2} T^{18} + 46552058051 p^{4} T^{20} + 818611682 p^{6} T^{22} + 17313788 p^{8} T^{24} + 339547 p^{10} T^{26} + 5770 p^{12} T^{28} + 113 p^{14} T^{30} + p^{16} T^{32} \)
59 \( ( 1 - 18 T + 314 T^{2} - 3708 T^{3} + 40525 T^{4} - 377055 T^{5} + 3414899 T^{6} - 27772695 T^{7} + 224517085 T^{8} - 27772695 p T^{9} + 3414899 p^{2} T^{10} - 377055 p^{3} T^{11} + 40525 p^{4} T^{12} - 3708 p^{5} T^{13} + 314 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
61 \( 1 - 335 T^{2} + 56959 T^{4} - 6951628 T^{6} + 698746562 T^{8} - 60376195337 T^{10} + 4597452492464 T^{12} - 317418120986212 T^{14} + 20167005200576653 T^{16} - 317418120986212 p^{2} T^{18} + 4597452492464 p^{4} T^{20} - 60376195337 p^{6} T^{22} + 698746562 p^{8} T^{24} - 6951628 p^{10} T^{26} + 56959 p^{12} T^{28} - 335 p^{14} T^{30} + p^{16} T^{32} \)
67 \( 1 + 291 T^{2} + 42453 T^{4} + 3563262 T^{6} + 173504320 T^{8} + 5808724449 T^{10} + 668928081240 T^{12} + 104142662733468 T^{14} + 9282721309480113 T^{16} + 104142662733468 p^{2} T^{18} + 668928081240 p^{4} T^{20} + 5808724449 p^{6} T^{22} + 173504320 p^{8} T^{24} + 3563262 p^{10} T^{26} + 42453 p^{12} T^{28} + 291 p^{14} T^{30} + p^{16} T^{32} \)
71 \( ( 1 + 3 T + 152 T^{2} + 99 T^{3} + 12195 T^{4} + 99 p T^{5} + 152 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
73 \( ( 1 + 12 T + 288 T^{2} + 2880 T^{3} + 41913 T^{4} + 394287 T^{5} + 4642215 T^{6} + 39318567 T^{7} + 394514699 T^{8} + 39318567 p T^{9} + 4642215 p^{2} T^{10} + 394287 p^{3} T^{11} + 41913 p^{4} T^{12} + 2880 p^{5} T^{13} + 288 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
79 \( 1 + 329 T^{2} + 48171 T^{4} + 4743088 T^{6} + 452223662 T^{8} + 46999974399 T^{10} + 4486961513032 T^{12} + 366158122457708 T^{14} + 28343208803731821 T^{16} + 366158122457708 p^{2} T^{18} + 4486961513032 p^{4} T^{20} + 46999974399 p^{6} T^{22} + 452223662 p^{8} T^{24} + 4743088 p^{10} T^{26} + 48171 p^{12} T^{28} + 329 p^{14} T^{30} + p^{16} T^{32} \)
83 \( ( 1 + 144 T^{2} + 7849 T^{4} + 1073880 T^{6} + 142819797 T^{8} + 1073880 p^{2} T^{10} + 7849 p^{4} T^{12} + 144 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( 1 - 189 T^{2} + 4064 T^{4} + 941031 T^{6} - 24268760 T^{8} + 3629642562 T^{10} - 1003871950027 T^{12} - 21285922063896 T^{14} + 12204608613267349 T^{16} - 21285922063896 p^{2} T^{18} - 1003871950027 p^{4} T^{20} + 3629642562 p^{6} T^{22} - 24268760 p^{8} T^{24} + 941031 p^{10} T^{26} + 4064 p^{12} T^{28} - 189 p^{14} T^{30} + p^{16} T^{32} \)
97 \( ( 1 + 320 T^{2} + 63357 T^{4} + 8594056 T^{6} + 944283317 T^{8} + 8594056 p^{2} T^{10} + 63357 p^{4} T^{12} + 320 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.15712055998557801224332102619, −3.08093743873416203250719509907, −2.89878969317962650576651263275, −2.86046675969596192243155773858, −2.83989972482543868767092710606, −2.81708122412169167576326979882, −2.72049842675075149488147407938, −2.61012244121069355594983670581, −2.37065379103877837370653107511, −2.32286186525318504779904020154, −2.23679816261749809671770551628, −2.20213415081372846328326812245, −2.13821039475195452286546717482, −1.88883282421863559594962965813, −1.75106627586566669586479221598, −1.64566438457182506840519986928, −1.63149096269540510989526483521, −1.53628269164787449815281827091, −1.24140427892923324241972167023, −1.03420836307792054662078013489, −0.984195589430343451700133147804, −0.907213411093585965160357886372, −0.816065964151386762035711075088, −0.43648854601517338674010776772, −0.01983197987722406832610642324, 0.01983197987722406832610642324, 0.43648854601517338674010776772, 0.816065964151386762035711075088, 0.907213411093585965160357886372, 0.984195589430343451700133147804, 1.03420836307792054662078013489, 1.24140427892923324241972167023, 1.53628269164787449815281827091, 1.63149096269540510989526483521, 1.64566438457182506840519986928, 1.75106627586566669586479221598, 1.88883282421863559594962965813, 2.13821039475195452286546717482, 2.20213415081372846328326812245, 2.23679816261749809671770551628, 2.32286186525318504779904020154, 2.37065379103877837370653107511, 2.61012244121069355594983670581, 2.72049842675075149488147407938, 2.81708122412169167576326979882, 2.83989972482543868767092710606, 2.86046675969596192243155773858, 2.89878969317962650576651263275, 3.08093743873416203250719509907, 3.15712055998557801224332102619

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.