Properties

Label 8-2352e4-1.1-c2e4-0-6
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $1.68690\times 10^{7}$
Root an. cond. $8.00545$
Motivic weight $2$
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  − 6·9-s + 30·11-s + 96·23-s − 11·25-s + 6·29-s − 146·37-s − 70·43-s − 198·53-s − 14·67-s + 204·71-s + 112·79-s + 27·81-s − 180·99-s + 174·107-s + 82·109-s − 432·113-s + 371·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 577·169-s + ⋯
L(s)  = 1  − 2/3·9-s + 2.72·11-s + 4.17·23-s − 0.439·25-s + 6/29·29-s − 3.94·37-s − 1.62·43-s − 3.73·53-s − 0.208·67-s + 2.87·71-s + 1.41·79-s + 1/3·81-s − 1.81·99-s + 1.62·107-s + 0.752·109-s − 3.82·113-s + 3.06·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.41·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.68690\times 10^{7}\)
Root analytic conductor: \(8.00545\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.181554635\)
\(L(\frac12)\) \(\approx\) \(2.181554635\)
\(L(2)\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
7 \( 1 \)
good5$D_4\times C_2$ \( 1 + 11 T^{2} - 36 T^{4} + 11 p^{4} T^{6} + p^{8} T^{8} \)
11$D_{4}$ \( ( 1 - 15 T + 152 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 577 T^{2} + 140208 T^{4} - 577 p^{4} T^{6} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 - 712 T^{2} + 272718 T^{4} - 712 p^{4} T^{6} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 - 253 T^{2} + 170028 T^{4} - 253 p^{4} T^{6} + p^{8} T^{8} \)
23$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )^{4} \)
29$D_{4}$ \( ( 1 - 3 T + 1538 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 50 T^{2} + 499827 T^{4} + 50 p^{4} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 + 73 T + 3924 T^{2} + 73 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 4300 T^{2} + 8926182 T^{4} - 4300 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 + 35 T + 2688 T^{2} + 35 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 7096 T^{2} + 21821166 T^{4} - 7096 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 + 99 T + 7922 T^{2} + 99 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 3985 T^{2} + 27245232 T^{4} - 3985 p^{4} T^{6} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 - 2500 T^{2} - 9084378 T^{4} - 2500 p^{4} T^{6} + p^{8} T^{8} \)
67$D_{4}$ \( ( 1 + 7 T + 5334 T^{2} + 7 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 - 102 T + 7418 T^{2} - 102 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 16201 T^{2} + 120802176 T^{4} - 16201 p^{4} T^{6} + p^{8} T^{8} \)
79$D_{4}$ \( ( 1 - 56 T + 12681 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 1283 T^{2} + 76688748 T^{4} + 1283 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 21640 T^{2} + 219622542 T^{4} - 21640 p^{4} T^{6} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 - 32425 T^{2} + 437832672 T^{4} - 32425 p^{4} T^{6} + p^{8} T^{8} \)
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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

−6.46169202032146553716569566593, −6.14734372291068335904887733186, −5.62337086005818864775771466617, −5.58330330054334043559700525707, −5.18886114873155789990219849164, −5.12369281884926042446297686524, −4.97171627564362993168882002685, −4.86957629712444937724315877341, −4.64040998926548491227648035484, −4.20283993645665554190671858539, −3.85895733242819000832047960484, −3.84978226035777102609205426336, −3.66207164186192474541234203791, −3.18907588121943278356054450798, −3.13525470112989965720783534382, −3.08717856019047749180231636666, −2.86346045338314995608482051424, −2.23609760803049206521442804129, −1.98840979755794598100597534865, −1.68069253572946154201350198612, −1.60693560935444820782152954596, −1.11155004996559074967842718405, −1.09235107499025842299297622627, −0.62987352809032726427524837208, −0.16643595394940157455100049632, 0.16643595394940157455100049632, 0.62987352809032726427524837208, 1.09235107499025842299297622627, 1.11155004996559074967842718405, 1.60693560935444820782152954596, 1.68069253572946154201350198612, 1.98840979755794598100597534865, 2.23609760803049206521442804129, 2.86346045338314995608482051424, 3.08717856019047749180231636666, 3.13525470112989965720783534382, 3.18907588121943278356054450798, 3.66207164186192474541234203791, 3.84978226035777102609205426336, 3.85895733242819000832047960484, 4.20283993645665554190671858539, 4.64040998926548491227648035484, 4.86957629712444937724315877341, 4.97171627564362993168882002685, 5.12369281884926042446297686524, 5.18886114873155789990219849164, 5.58330330054334043559700525707, 5.62337086005818864775771466617, 6.14734372291068335904887733186, 6.46169202032146553716569566593

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