Properties

Label 8-252e4-1.1-c3e4-0-3
Degree $8$
Conductor $4032758016$
Sign $1$
Analytic cond. $48872.7$
Root an. cond. $3.85596$
Motivic weight $3$
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  + 11·5-s + 6·7-s − 5·11-s + 10·13-s + 100·17-s − 67·19-s + 76·23-s + 232·25-s − 550·29-s − 362·31-s + 66·35-s + 5·37-s + 324·41-s + 1.44e3·43-s − 216·47-s + 113·49-s + 495·53-s − 55·55-s + 173·59-s + 532·61-s + 110·65-s − 111·67-s − 3.20e3·71-s − 1.21e3·73-s − 30·77-s − 1.46e3·79-s + 2.81e3·83-s + ⋯
L(s)  = 1  + 0.983·5-s + 0.323·7-s − 0.137·11-s + 0.213·13-s + 1.42·17-s − 0.808·19-s + 0.689·23-s + 1.85·25-s − 3.52·29-s − 2.09·31-s + 0.318·35-s + 0.0222·37-s + 1.23·41-s + 5.11·43-s − 0.670·47-s + 0.329·49-s + 1.28·53-s − 0.134·55-s + 0.381·59-s + 1.11·61-s + 0.209·65-s − 0.202·67-s − 5.34·71-s − 1.94·73-s − 0.0444·77-s − 2.07·79-s + 3.72·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(48872.7\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{252} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(4.520505165\)
\(L(\frac12)\) \(\approx\) \(4.520505165\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7$C_2^2$ \( 1 - 6 T - 11 p T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
good5$D_4\times C_2$ \( 1 - 11 T - 111 T^{2} + 198 T^{3} + 23074 T^{4} + 198 p^{3} T^{5} - 111 p^{6} T^{6} - 11 p^{9} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 + 5 T - 279 T^{2} - 11790 T^{3} - 1712420 T^{4} - 11790 p^{3} T^{5} - 279 p^{6} T^{6} + 5 p^{9} T^{7} + p^{12} T^{8} \)
13$D_{4}$ \( ( 1 - 5 T + 3194 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 100 T - 1554 T^{2} - 172800 T^{3} + 60227347 T^{4} - 172800 p^{3} T^{5} - 1554 p^{6} T^{6} - 100 p^{9} T^{7} + p^{12} T^{8} \)
19$D_4\times C_2$ \( 1 + 67 T - 9917 T^{2} + 46096 T^{3} + 129696904 T^{4} + 46096 p^{3} T^{5} - 9917 p^{6} T^{6} + 67 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 - 76 T - 702 T^{2} + 1357056 T^{3} - 176347997 T^{4} + 1357056 p^{3} T^{5} - 702 p^{6} T^{6} - 76 p^{9} T^{7} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 + 275 T + 61846 T^{2} + 275 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 362 T + 1871 p T^{2} + 4872882 T^{3} + 543844364 T^{4} + 4872882 p^{3} T^{5} + 1871 p^{7} T^{6} + 362 p^{9} T^{7} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 - 5 T - 3581 T^{2} + 488500 T^{3} - 2553989498 T^{4} + 488500 p^{3} T^{5} - 3581 p^{6} T^{6} - 5 p^{9} T^{7} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 - 162 T + 128770 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 721 T + 288540 T^{2} - 721 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 216 T - 110122 T^{2} - 10987488 T^{3} + 8956160067 T^{4} - 10987488 p^{3} T^{5} - 110122 p^{6} T^{6} + 216 p^{9} T^{7} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 495 T - 110077 T^{2} - 28387260 T^{3} + 67454482350 T^{4} - 28387260 p^{3} T^{5} - 110077 p^{6} T^{6} - 495 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 - 173 T - 252777 T^{2} + 22152996 T^{3} + 31595360704 T^{4} + 22152996 p^{3} T^{5} - 252777 p^{6} T^{6} - 173 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 - 532 T - 164494 T^{2} + 3428208 T^{3} + 84510915419 T^{4} + 3428208 p^{3} T^{5} - 164494 p^{6} T^{6} - 532 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 + 111 T - 306211 T^{2} - 31412334 T^{3} + 7298551932 T^{4} - 31412334 p^{3} T^{5} - 306211 p^{6} T^{6} + 111 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 + 1600 T + 1262410 T^{2} + 1600 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 1215 T + 464669 T^{2} + 283729230 T^{3} + 297634694022 T^{4} + 283729230 p^{3} T^{5} + 464669 p^{6} T^{6} + 1215 p^{9} T^{7} + p^{12} T^{8} \)
79$D_4\times C_2$ \( 1 + 1460 T + 798095 T^{2} + 507243420 T^{3} + 484186197104 T^{4} + 507243420 p^{3} T^{5} + 798095 p^{6} T^{6} + 1460 p^{9} T^{7} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 - 1409 T + 1147696 T^{2} - 1409 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 1974 T + 1597682 T^{2} + 1754996544 T^{3} + 2041356338031 T^{4} + 1754996544 p^{3} T^{5} + 1597682 p^{6} T^{6} + 1974 p^{9} T^{7} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 - 561 T + 1863448 T^{2} - 561 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−8.601550147966004062059628429490, −7.79280165363481422805380170395, −7.69466378693065326644719018936, −7.54505550077357687680409810046, −7.28446770746705004066564893198, −7.05123639630628479690641256614, −6.98974747886539780599569687223, −6.19201184052812069689139503759, −5.95478211976424277168533670507, −5.87154697919865649606555660428, −5.61497349130017592233574578158, −5.46756351735909477918216611409, −5.23278970930816925473995755337, −4.47693625995734026485981832455, −4.42307114227122256890539326728, −4.12986023873278750563991157004, −3.69692071881759325259472680495, −3.38630150627520310377486275048, −2.94317872453654372056680375886, −2.46782145476360126172185158771, −2.36621514303353230891687028253, −1.59706414907624448631573874254, −1.56766565591330115095556364780, −0.899653352061410444339938121024, −0.40455580058650805894299539063, 0.40455580058650805894299539063, 0.899653352061410444339938121024, 1.56766565591330115095556364780, 1.59706414907624448631573874254, 2.36621514303353230891687028253, 2.46782145476360126172185158771, 2.94317872453654372056680375886, 3.38630150627520310377486275048, 3.69692071881759325259472680495, 4.12986023873278750563991157004, 4.42307114227122256890539326728, 4.47693625995734026485981832455, 5.23278970930816925473995755337, 5.46756351735909477918216611409, 5.61497349130017592233574578158, 5.87154697919865649606555660428, 5.95478211976424277168533670507, 6.19201184052812069689139503759, 6.98974747886539780599569687223, 7.05123639630628479690641256614, 7.28446770746705004066564893198, 7.54505550077357687680409810046, 7.69466378693065326644719018936, 7.79280165363481422805380170395, 8.601550147966004062059628429490

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