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Properties

Label 32-552e16-1.1-c1e16-0-0
Degree $32$
Conductor $7.431\times 10^{43}$
Sign $1$
Analytic cond. $2.02982\times 10^{10}$
Root an. cond. $2.09946$
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  + 4·3-s + 10·9-s + 8·13-s − 24·25-s + 24·27-s + 56·31-s + 32·39-s + 40·49-s + 40·73-s − 96·75-s + 51·81-s + 224·93-s + 80·117-s − 80·121-s + 127-s + 131-s + 137-s + 139-s + 160·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 28·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2.30·3-s + 10/3·9-s + 2.21·13-s − 4.79·25-s + 4.61·27-s + 10.0·31-s + 5.12·39-s + 40/7·49-s + 4.68·73-s − 11.0·75-s + 17/3·81-s + 23.2·93-s + 7.39·117-s − 7.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 13.1·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.15·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{48} \cdot 3^{16} \cdot 23^{16}\)
Sign: $1$
Analytic conductor: \(2.02982\times 10^{10}\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{48} \cdot 3^{16} \cdot 23^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.064363289\)
\(L(\frac12)\) \(\approx\) \(5.064363289\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 - 2 T + T^{2} - 2 T^{3} + 4 T^{4} - 2 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 8 T^{2} - 164 T^{4} + 4680 T^{6} + 158726 T^{8} + 4680 p^{2} T^{10} - 164 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \)
good5 \( ( 1 + 12 T^{2} + 108 T^{4} + 756 T^{6} + 4038 T^{8} + 756 p^{2} T^{10} + 108 p^{4} T^{12} + 12 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
7 \( ( 1 - 20 T^{2} + 300 T^{4} - 2956 T^{6} + 24102 T^{8} - 2956 p^{2} T^{10} + 300 p^{4} T^{12} - 20 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
11 \( ( 1 + 40 T^{2} + 940 T^{4} + 15768 T^{6} + 198662 T^{8} + 15768 p^{2} T^{10} + 940 p^{4} T^{12} + 40 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
13 \( ( 1 - 2 T + 17 T^{2} + 66 T^{3} - 36 T^{4} + 66 p T^{5} + 17 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
17 \( ( 1 + 108 T^{2} + 5484 T^{4} + 169908 T^{6} + 3501414 T^{8} + 169908 p^{2} T^{10} + 5484 p^{4} T^{12} + 108 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( ( 1 - 84 T^{2} + 3868 T^{4} - 118508 T^{6} + 2635046 T^{8} - 118508 p^{2} T^{10} + 3868 p^{4} T^{12} - 84 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
29 \( ( 1 - 162 T^{2} + 12689 T^{4} - 630306 T^{6} + 21708724 T^{8} - 630306 p^{2} T^{10} + 12689 p^{4} T^{12} - 162 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 - 14 T + 185 T^{2} - 1390 T^{3} + 9580 T^{4} - 1390 p T^{5} + 185 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
37 \( ( 1 - 120 T^{2} + 7148 T^{4} - 317384 T^{6} + 12423942 T^{8} - 317384 p^{2} T^{10} + 7148 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
41 \( ( 1 - 242 T^{2} + 27825 T^{4} - 1996882 T^{6} + 97903940 T^{8} - 1996882 p^{2} T^{10} + 27825 p^{4} T^{12} - 242 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 - 116 T^{2} + 4380 T^{4} - 9100 T^{6} - 3773850 T^{8} - 9100 p^{2} T^{10} + 4380 p^{4} T^{12} - 116 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
47 \( ( 1 - 274 T^{2} + 34577 T^{4} - 2708866 T^{6} + 149008260 T^{8} - 2708866 p^{2} T^{10} + 34577 p^{4} T^{12} - 274 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 + 76 T^{2} + 7260 T^{4} + 543668 T^{6} + 26967270 T^{8} + 543668 p^{2} T^{10} + 7260 p^{4} T^{12} + 76 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
59 \( ( 1 - 248 T^{2} + 26716 T^{4} - 1720584 T^{6} + 94978982 T^{8} - 1720584 p^{2} T^{10} + 26716 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 + 8 T^{2} + 140 p T^{4} + 58232 T^{6} + 45113190 T^{8} + 58232 p^{2} T^{10} + 140 p^{5} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
67 \( ( 1 - 244 T^{2} + 40236 T^{4} - 4188236 T^{6} + 4994178 p T^{8} - 4188236 p^{2} T^{10} + 40236 p^{4} T^{12} - 244 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
71 \( ( 1 - 258 T^{2} + 39025 T^{4} - 4161602 T^{6} + 333173252 T^{8} - 4161602 p^{2} T^{10} + 39025 p^{4} T^{12} - 258 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 - 10 T + 217 T^{2} - 1434 T^{3} + 20516 T^{4} - 1434 p T^{5} + 217 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
79 \( ( 1 - 212 T^{2} + 31388 T^{4} - 3438732 T^{6} + 291876486 T^{8} - 3438732 p^{2} T^{10} + 31388 p^{4} T^{12} - 212 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
83 \( ( 1 + 40 T^{2} + 14444 T^{4} + 847896 T^{6} + 125156934 T^{8} + 847896 p^{2} T^{10} + 14444 p^{4} T^{12} + 40 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( ( 1 + 140 T^{2} + 19356 T^{4} + 1230676 T^{6} + 131424966 T^{8} + 1230676 p^{2} T^{10} + 19356 p^{4} T^{12} + 140 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 - 200 T^{2} + 27884 T^{4} - 2015928 T^{6} + 180339750 T^{8} - 2015928 p^{2} T^{10} + 27884 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
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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

−2.81148492566398160824289905008, −2.76327880060513217857903161352, −2.69785554029410178483168830898, −2.62190522430456357811593211153, −2.60100469486453286595793139435, −2.56260062031666796235407111387, −2.38293118986593074827462824062, −2.35322125469497166030226397059, −2.29387220928648367323578017732, −2.29260461430445556177432491355, −2.15040302867546888489592198551, −2.05950784651006590609647526155, −2.03309941461137870412469221080, −1.68754224913395606375613162319, −1.46436778415693328027134450368, −1.44444289581607143014595289540, −1.36145812268064785181888150100, −1.32992934454986032771565915747, −1.18017307149640458402427395063, −1.10828824178639643952328119518, −1.02697618451288132871049900633, −0.863765898455623397519133123994, −0.797474083992520064441210314325, −0.54217477248058765506592308159, −0.07783677714361577774562670036, 0.07783677714361577774562670036, 0.54217477248058765506592308159, 0.797474083992520064441210314325, 0.863765898455623397519133123994, 1.02697618451288132871049900633, 1.10828824178639643952328119518, 1.18017307149640458402427395063, 1.32992934454986032771565915747, 1.36145812268064785181888150100, 1.44444289581607143014595289540, 1.46436778415693328027134450368, 1.68754224913395606375613162319, 2.03309941461137870412469221080, 2.05950784651006590609647526155, 2.15040302867546888489592198551, 2.29260461430445556177432491355, 2.29387220928648367323578017732, 2.35322125469497166030226397059, 2.38293118986593074827462824062, 2.56260062031666796235407111387, 2.60100469486453286595793139435, 2.62190522430456357811593211153, 2.69785554029410178483168830898, 2.76327880060513217857903161352, 2.81148492566398160824289905008

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

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