Properties

Label 16-2352e8-1.1-c1e8-0-14
Degree $16$
Conductor $9.365\times 10^{26}$
Sign $1$
Analytic cond. $1.54780\times 10^{10}$
Root an. cond. $4.33368$
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  + 8·3-s + 36·9-s + 8·25-s + 120·27-s + 16·29-s − 32·31-s − 16·47-s + 16·53-s + 48·59-s + 64·75-s + 330·81-s + 128·87-s − 256·93-s + 16·109-s + 48·121-s + 127-s + 131-s + 137-s + 139-s − 128·141-s + 149-s + 151-s + 157-s + 128·159-s + 163-s + 167-s + 64·169-s + ⋯
L(s)  = 1  + 4.61·3-s + 12·9-s + 8/5·25-s + 23.0·27-s + 2.97·29-s − 5.74·31-s − 2.33·47-s + 2.19·53-s + 6.24·59-s + 7.39·75-s + 36.6·81-s + 13.7·87-s − 26.5·93-s + 1.53·109-s + 4.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.7·141-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 10.1·159-s + 0.0783·163-s + 0.0773·167-s + 4.92·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{8} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(1.54780\times 10^{10}\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{8} \cdot 7^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(170.3694864\)
\(L(\frac12)\) \(\approx\) \(170.3694864\)
\(L(\frac{3}{2})\) 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 \( ( 1 - T )^{8} \)
7 \( 1 \)
good5 \( 1 - 8 T^{2} + 24 T^{4} - 8 p T^{6} + 146 T^{8} - 8 p^{3} T^{10} + 24 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 - 48 T^{2} + 1140 T^{4} - 18576 T^{6} + 231302 T^{8} - 18576 p^{2} T^{10} + 1140 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 - 32 T^{2} + 592 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( 1 - 72 T^{2} + 2424 T^{4} - 52008 T^{6} + 914354 T^{8} - 52008 p^{2} T^{10} + 2424 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + 36 T^{2} + 96 T^{3} + 590 T^{4} + 96 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( 1 - 96 T^{2} + 4980 T^{4} - 175392 T^{6} + 4613990 T^{8} - 175392 p^{2} T^{10} + 4980 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 - 8 T + 112 T^{2} - 664 T^{3} + 4842 T^{4} - 664 p T^{5} + 112 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 16 T + 180 T^{2} + 1328 T^{3} + 8414 T^{4} + 1328 p T^{5} + 180 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 48 T^{2} + 192 T^{3} + 1778 T^{4} + 192 p T^{5} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
41 \( 1 - 248 T^{2} + 29208 T^{4} - 2129944 T^{6} + 2560930 p T^{8} - 2129944 p^{2} T^{10} + 29208 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 + 24 T^{2} - 324 T^{4} + 63336 T^{6} + 5311334 T^{8} + 63336 p^{2} T^{10} - 324 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 + 8 T + 172 T^{2} + 904 T^{3} + 11358 T^{4} + 904 p T^{5} + 172 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 8 T + 124 T^{2} - 1240 T^{3} + 8310 T^{4} - 1240 p T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( ( 1 - 24 T + 316 T^{2} - 2808 T^{3} + 21870 T^{4} - 2808 p T^{5} + 316 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( 1 - 192 T^{2} + 19488 T^{4} - 1664448 T^{6} + 117511586 T^{8} - 1664448 p^{2} T^{10} + 19488 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 280 T^{2} + 40956 T^{4} - 4148456 T^{6} + 317117990 T^{8} - 4148456 p^{2} T^{10} + 40956 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 - 208 T^{2} + 27604 T^{4} - 2815216 T^{6} + 229790374 T^{8} - 2815216 p^{2} T^{10} + 27604 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 - 256 T^{2} + 38784 T^{4} - 4262912 T^{6} + 356271938 T^{8} - 4262912 p^{2} T^{10} + 38784 p^{4} T^{12} - 256 p^{6} T^{14} + p^{8} T^{16} \)
79 \( 1 - 328 T^{2} + 55068 T^{4} - 6274040 T^{6} + 549903302 T^{8} - 6274040 p^{2} T^{10} + 55068 p^{4} T^{12} - 328 p^{6} T^{14} + p^{8} T^{16} \)
83 \( ( 1 + 76 T^{2} - 192 T^{3} + 11094 T^{4} - 192 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( 1 + 120 T^{2} + 34680 T^{4} + 2809368 T^{6} + 422712050 T^{8} + 2809368 p^{2} T^{10} + 34680 p^{4} T^{12} + 120 p^{6} T^{14} + p^{8} T^{16} \)
97 \( 1 - 192 T^{2} + 44928 T^{4} - 5229504 T^{6} + 668134658 T^{8} - 5229504 p^{2} T^{10} + 44928 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.67997388518629920880349338206, −3.59538953167799741886727697609, −3.57766255148176313338063004705, −3.29886500293680413260623788443, −3.24963041328052264143719738744, −3.12568087775161857595044735065, −3.04567783545437821169989824999, −2.97441852000047671179891596844, −2.68800276432933632743689611105, −2.58469725817770469206713267874, −2.54832094054360471954389346382, −2.46164978671375085609255082186, −2.31505538536968367248004785318, −2.25078439029949990646889721133, −1.94738014767813903280436399856, −1.80797982860474194924560574346, −1.80426289732509973409288503985, −1.71158374473677723181529034224, −1.51245503948935770875956896603, −1.42996832965711412307643694087, −1.12954921798893029611466155470, −0.855722908512736566870775507038, −0.64347142892233855644111219742, −0.63198979858857738614498090285, −0.42514051841251673791421299740, 0.42514051841251673791421299740, 0.63198979858857738614498090285, 0.64347142892233855644111219742, 0.855722908512736566870775507038, 1.12954921798893029611466155470, 1.42996832965711412307643694087, 1.51245503948935770875956896603, 1.71158374473677723181529034224, 1.80426289732509973409288503985, 1.80797982860474194924560574346, 1.94738014767813903280436399856, 2.25078439029949990646889721133, 2.31505538536968367248004785318, 2.46164978671375085609255082186, 2.54832094054360471954389346382, 2.58469725817770469206713267874, 2.68800276432933632743689611105, 2.97441852000047671179891596844, 3.04567783545437821169989824999, 3.12568087775161857595044735065, 3.24963041328052264143719738744, 3.29886500293680413260623788443, 3.57766255148176313338063004705, 3.59538953167799741886727697609, 3.67997388518629920880349338206

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

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