Properties

Label 8-2475e4-1.1-c1e4-0-12
Degree $8$
Conductor $3.752\times 10^{13}$
Sign $1$
Analytic cond. $152548.$
Root an. cond. $4.44555$
Motivic weight $1$
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  − 2·2-s + 2·4-s − 4·7-s − 2·8-s − 4·11-s − 8·13-s + 8·14-s + 16-s − 4·17-s + 4·19-s + 8·22-s + 8·23-s + 16·26-s − 8·28-s − 4·29-s + 8·34-s − 8·37-s − 8·38-s − 4·41-s − 12·43-s − 8·44-s − 16·46-s + 4·49-s − 16·52-s − 16·53-s + 8·56-s + 8·58-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.51·7-s − 0.707·8-s − 1.20·11-s − 2.21·13-s + 2.13·14-s + 1/4·16-s − 0.970·17-s + 0.917·19-s + 1.70·22-s + 1.66·23-s + 3.13·26-s − 1.51·28-s − 0.742·29-s + 1.37·34-s − 1.31·37-s − 1.29·38-s − 0.624·41-s − 1.82·43-s − 1.20·44-s − 2.35·46-s + 4/7·49-s − 2.21·52-s − 2.19·53-s + 1.06·56-s + 1.05·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(152548.\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{8} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{4} \)
good2$C_2 \wr S_4$ \( 1 + p T + p T^{2} + p T^{3} + 3 T^{4} + p^{2} T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 4 T + 12 T^{2} + 20 T^{3} + 34 T^{4} + 20 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 8 T + 44 T^{2} + 144 T^{3} + 514 T^{4} + 144 p T^{5} + 44 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 4 T + 28 T^{2} - 36 T^{3} + 50 T^{4} - 36 p T^{5} + 28 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 4 T + 20 T^{2} - 36 T^{3} + 326 T^{4} - 36 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 8 T + 60 T^{2} - 296 T^{3} + 1510 T^{4} - 296 p T^{5} + 60 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 4 T + 28 T^{2} + 108 T^{3} - 202 T^{4} + 108 p T^{5} + 28 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 36 T^{2} + 192 T^{3} + 454 T^{4} + 192 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 8 T + 92 T^{2} + 664 T^{3} + 5046 T^{4} + 664 p T^{5} + 92 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 4 T + 44 T^{2} + 60 T^{3} + 2406 T^{4} + 60 p T^{5} + 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 12 T + 204 T^{2} + 36 p T^{3} + 13810 T^{4} + 36 p^{2} T^{5} + 204 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 60 T^{2} - 576 T^{3} + 646 T^{4} - 576 p T^{5} + 60 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 16 T + 252 T^{2} + 2416 T^{3} + 20854 T^{4} + 2416 p T^{5} + 252 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 24 T + 324 T^{2} + 2872 T^{3} + 386 p T^{4} + 2872 p T^{5} + 324 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 8 T + 140 T^{2} - 408 T^{3} + 7414 T^{4} - 408 p T^{5} + 140 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 44 T^{2} + 576 T^{3} + 3126 T^{4} + 576 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 16 T + 324 T^{2} + 3280 T^{3} + 35686 T^{4} + 3280 p T^{5} + 324 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 8 T + 284 T^{2} + 1584 T^{3} + 418 p T^{4} + 1584 p T^{5} + 284 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 12 T + 308 T^{2} - 2652 T^{3} + 36342 T^{4} - 2652 p T^{5} + 308 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 8 T + 260 T^{2} + 1632 T^{3} + 31218 T^{4} + 1632 p T^{5} + 260 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 16 T + 332 T^{2} + 3760 T^{3} + 43974 T^{4} + 3760 p T^{5} + 332 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 8 T + 284 T^{2} + 1144 T^{3} + 33414 T^{4} + 1144 p T^{5} + 284 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.78135750080551938098598038225, −6.67862481449905231150754071024, −6.35745907240403012874933713275, −6.23134715515912704447210184323, −6.03668413142073307492831184747, −5.81353256317024106698026910451, −5.49279896362091239205037751011, −5.18577539998379245190015316238, −5.17435191645933910901506171867, −4.80963572042033586330958311639, −4.79008084436786905946865223636, −4.58705779911231931167183707913, −4.40079222477721316422423056995, −3.84732777938256627157866436143, −3.49751291747361428497418928534, −3.46661776723087736976729383866, −3.20328313525948568469163226769, −3.04524970294702307493917238860, −2.70076868064484784101297123966, −2.54839554962171058184658256008, −2.33318510891204816072559424223, −2.07676708373068819871034597401, −1.48017990563307550638536501844, −1.40569169728597710718909065005, −1.13947779835390337591176701637, 0, 0, 0, 0, 1.13947779835390337591176701637, 1.40569169728597710718909065005, 1.48017990563307550638536501844, 2.07676708373068819871034597401, 2.33318510891204816072559424223, 2.54839554962171058184658256008, 2.70076868064484784101297123966, 3.04524970294702307493917238860, 3.20328313525948568469163226769, 3.46661776723087736976729383866, 3.49751291747361428497418928534, 3.84732777938256627157866436143, 4.40079222477721316422423056995, 4.58705779911231931167183707913, 4.79008084436786905946865223636, 4.80963572042033586330958311639, 5.17435191645933910901506171867, 5.18577539998379245190015316238, 5.49279896362091239205037751011, 5.81353256317024106698026910451, 6.03668413142073307492831184747, 6.23134715515912704447210184323, 6.35745907240403012874933713275, 6.67862481449905231150754071024, 6.78135750080551938098598038225

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