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Properties

Label 8-912e4-1.1-c3e4-0-3
Degree $8$
Conductor $691798081536$
Sign $1$
Analytic cond. $8.38385\times 10^{6}$
Root an. cond. $7.33551$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·3-s − 6·5-s − 38·7-s + 90·9-s + 68·13-s + 72·15-s + 30·17-s − 76·19-s + 456·21-s + 246·23-s − 173·25-s − 540·27-s + 6·29-s − 188·31-s + 228·35-s + 620·37-s − 816·39-s − 294·41-s − 218·43-s − 540·45-s − 672·47-s + 303·49-s − 360·51-s + 294·53-s + 912·57-s − 588·59-s − 514·61-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.536·5-s − 2.05·7-s + 10/3·9-s + 1.45·13-s + 1.23·15-s + 0.428·17-s − 0.917·19-s + 4.73·21-s + 2.23·23-s − 1.38·25-s − 3.84·27-s + 0.0384·29-s − 1.08·31-s + 1.10·35-s + 2.75·37-s − 3.35·39-s − 1.11·41-s − 0.773·43-s − 1.78·45-s − 2.08·47-s + 0.883·49-s − 0.988·51-s + 0.761·53-s + 2.11·57-s − 1.29·59-s − 1.07·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(8.38385\times 10^{6}\)
Root analytic conductor: \(7.33551\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 19^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + p T )^{4} \)
19$C_1$ \( ( 1 + p T )^{4} \)
good5$C_2 \wr S_4$ \( 1 + 6 T + 209 T^{2} + 1578 T^{3} + 40596 T^{4} + 1578 p^{3} T^{5} + 209 p^{6} T^{6} + 6 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 38 T + 163 p T^{2} + 21662 T^{3} + 461092 T^{4} + 21662 p^{3} T^{5} + 163 p^{7} T^{6} + 38 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 1769 T^{2} - 56106 T^{3} + 1777452 T^{4} - 56106 p^{3} T^{5} + 1769 p^{6} T^{6} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 68 T + 6580 T^{2} - 379820 T^{3} + 20707462 T^{4} - 379820 p^{3} T^{5} + 6580 p^{6} T^{6} - 68 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 30 T + 12377 T^{2} - 358170 T^{3} + 85588764 T^{4} - 358170 p^{3} T^{5} + 12377 p^{6} T^{6} - 30 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 246 T + 57020 T^{2} - 8349198 T^{3} + 1068194790 T^{4} - 8349198 p^{3} T^{5} + 57020 p^{6} T^{6} - 246 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 6 T + 60728 T^{2} + 160350 T^{3} + 2039540382 T^{4} + 160350 p^{3} T^{5} + 60728 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 188 T + 112480 T^{2} + 15450668 T^{3} + 4952892094 T^{4} + 15450668 p^{3} T^{5} + 112480 p^{6} T^{6} + 188 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 620 T + 213496 T^{2} - 58159988 T^{3} + 13588894366 T^{4} - 58159988 p^{3} T^{5} + 213496 p^{6} T^{6} - 620 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 294 T + 111716 T^{2} + 27159906 T^{3} + 11729887014 T^{4} + 27159906 p^{3} T^{5} + 111716 p^{6} T^{6} + 294 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 218 T + 83413 T^{2} + 1457750 T^{3} + 4916444908 T^{4} + 1457750 p^{3} T^{5} + 83413 p^{6} T^{6} + 218 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 672 T + 450449 T^{2} + 175276650 T^{3} + 69429847404 T^{4} + 175276650 p^{3} T^{5} + 450449 p^{6} T^{6} + 672 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 294 T + 306776 T^{2} - 122723634 T^{3} + 55572584190 T^{4} - 122723634 p^{3} T^{5} + 306776 p^{6} T^{6} - 294 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 588 T + 541196 T^{2} + 176844588 T^{3} + 122576127798 T^{4} + 176844588 p^{3} T^{5} + 541196 p^{6} T^{6} + 588 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 514 T + 781345 T^{2} + 317906854 T^{3} + 253526672788 T^{4} + 317906854 p^{3} T^{5} + 781345 p^{6} T^{6} + 514 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 368 T + 429148 T^{2} + 234238448 T^{3} + 185941419190 T^{4} + 234238448 p^{3} T^{5} + 429148 p^{6} T^{6} + 368 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 516 T + 428108 T^{2} + 288993876 T^{3} + 144430140294 T^{4} + 288993876 p^{3} T^{5} + 428108 p^{6} T^{6} + 516 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 166 T + 574489 T^{2} - 293085254 T^{3} + 82506112156 T^{4} - 293085254 p^{3} T^{5} + 574489 p^{6} T^{6} + 166 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 616 T + 1425628 T^{2} - 784934728 T^{3} + 976025833606 T^{4} - 784934728 p^{3} T^{5} + 1425628 p^{6} T^{6} - 616 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 1062 T + 619412 T^{2} - 464593158 T^{3} + 581397065286 T^{4} - 464593158 p^{3} T^{5} + 619412 p^{6} T^{6} - 1062 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 3162 T + 5987648 T^{2} + 7512056502 T^{3} + 7267150013982 T^{4} + 7512056502 p^{3} T^{5} + 5987648 p^{6} T^{6} + 3162 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 2188 T + 4656436 T^{2} + 5915759860 T^{3} + 6760606056982 T^{4} + 5915759860 p^{3} T^{5} + 4656436 p^{6} T^{6} + 2188 p^{9} T^{7} + p^{12} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.26695760294553263748212421464, −6.78400391396245586950962219197, −6.62479121197126616504805710122, −6.59083705094933470855261696816, −6.48060073315327317055184567672, −6.01319121347108333204938557622, −6.00428677880379393800214387202, −5.87271592913972666143254555193, −5.60114714215580358549073490010, −5.09651869925016712621805336687, −5.04300939650957196477796840073, −4.84251742579803117013279641485, −4.66200616888459121890917493754, −4.06674513970246949572642174271, −3.85038228420457432826690060388, −3.79212111639829442536386486928, −3.78187162949345077116569979735, −3.12830757245974763797552242517, −2.92755920679588106597087077257, −2.71683219976730441059981837796, −2.38672472517898946607538392034, −1.60702345582993282701844725235, −1.39940630250291071575168508880, −1.18067133560018851161415438783, −1.05439312461156566372113736280, 0, 0, 0, 0, 1.05439312461156566372113736280, 1.18067133560018851161415438783, 1.39940630250291071575168508880, 1.60702345582993282701844725235, 2.38672472517898946607538392034, 2.71683219976730441059981837796, 2.92755920679588106597087077257, 3.12830757245974763797552242517, 3.78187162949345077116569979735, 3.79212111639829442536386486928, 3.85038228420457432826690060388, 4.06674513970246949572642174271, 4.66200616888459121890917493754, 4.84251742579803117013279641485, 5.04300939650957196477796840073, 5.09651869925016712621805336687, 5.60114714215580358549073490010, 5.87271592913972666143254555193, 6.00428677880379393800214387202, 6.01319121347108333204938557622, 6.48060073315327317055184567672, 6.59083705094933470855261696816, 6.62479121197126616504805710122, 6.78400391396245586950962219197, 7.26695760294553263748212421464

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