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Properties

Label 8-475e4-1.1-c1e4-0-5
Degree $8$
Conductor $50906640625$
Sign $1$
Analytic cond. $206.958$
Root an. cond. $1.94753$
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  + 2·2-s − 2·3-s + 2·4-s − 4·6-s − 4·7-s + 4·8-s + 4·11-s − 4·12-s − 2·13-s − 8·14-s + 3·16-s − 4·17-s + 4·19-s + 8·21-s + 8·22-s + 8·23-s − 8·24-s − 4·26-s + 6·27-s − 8·28-s + 4·29-s + 4·31-s − 4·32-s − 8·33-s − 8·34-s + 6·37-s + 8·38-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 4-s − 1.63·6-s − 1.51·7-s + 1.41·8-s + 1.20·11-s − 1.15·12-s − 0.554·13-s − 2.13·14-s + 3/4·16-s − 0.970·17-s + 0.917·19-s + 1.74·21-s + 1.70·22-s + 1.66·23-s − 1.63·24-s − 0.784·26-s + 1.15·27-s − 1.51·28-s + 0.742·29-s + 0.718·31-s − 0.707·32-s − 1.39·33-s − 1.37·34-s + 0.986·37-s + 1.29·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(206.958\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.050871497\)
\(L(\frac12)\) \(\approx\) \(3.050871497\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
19$C_1$ \( ( 1 - T )^{4} \)
good2$C_2 \wr S_4$ \( 1 - p T + p T^{2} - p^{2} T^{3} + 9 T^{4} - p^{3} T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
3$C_2 \wr S_4$ \( 1 + 2 T + 4 T^{2} + 2 T^{3} + 2 T^{4} + 2 p T^{5} + 4 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
7$S_4\times C_2$ \( 1 + 4 T + 12 T^{2} + 36 T^{3} + 102 T^{4} + 36 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$\textrm{GL(2,3)}$ \( 1 - 4 T + 28 T^{2} - 100 T^{3} + 422 T^{4} - 100 p T^{5} + 28 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 2 T + 28 T^{2} + 46 T^{3} + 410 T^{4} + 46 p T^{5} + 28 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 4 T + 36 T^{2} + 188 T^{3} + 694 T^{4} + 188 p T^{5} + 36 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 8 T + 68 T^{2} - 376 T^{3} + 2358 T^{4} - 376 p T^{5} + 68 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
29$\textrm{GL(2,3)}$ \( 1 - 4 T + 84 T^{2} - 332 T^{3} + 3238 T^{4} - 332 p T^{5} + 84 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 4 T + 44 T^{2} + 140 T^{3} + 166 T^{4} + 140 p T^{5} + 44 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 6 T + 124 T^{2} - 626 T^{3} + 6442 T^{4} - 626 p T^{5} + 124 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 16 T + 220 T^{2} - 1936 T^{3} + 14438 T^{4} - 1936 p T^{5} + 220 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 4 T + 156 T^{2} + 468 T^{3} + 9750 T^{4} + 468 p T^{5} + 156 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 12 T + 124 T^{2} - 1036 T^{3} + 8294 T^{4} - 1036 p T^{5} + 124 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 10 T + 4 p T^{2} - 1406 T^{3} + 16506 T^{4} - 1406 p T^{5} + 4 p^{3} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 172 T^{2} - 224 T^{3} + 13142 T^{4} - 224 p T^{5} + 172 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 20 T + 300 T^{2} - 2972 T^{3} + 26502 T^{4} - 2972 p T^{5} + 300 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 18 T + 276 T^{2} - 3130 T^{3} + 26930 T^{4} - 3130 p T^{5} + 276 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 20 T + 316 T^{2} + 3236 T^{3} + 30566 T^{4} + 3236 p T^{5} + 316 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 28 T + 548 T^{2} + 6916 T^{3} + 69526 T^{4} + 6916 p T^{5} + 548 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 16 T + 348 T^{2} + 3312 T^{3} + 40646 T^{4} + 3312 p T^{5} + 348 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 260 T^{2} + 112 T^{3} + 29862 T^{4} + 112 p T^{5} + 260 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 4 T + 212 T^{2} - 1244 T^{3} + 22134 T^{4} - 1244 p T^{5} + 212 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 30 T + 612 T^{2} + 8738 T^{3} + 98522 T^{4} + 8738 p T^{5} + 612 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.62731377401260901881907374390, −7.50887323311184825290869264057, −7.37293913524197533799493645969, −7.00563683171543417120531968409, −6.81889230570599300259343450750, −6.76894716193023402380030785973, −6.49472017848852510188881082873, −6.13384615958499342549676490383, −5.94629681735106961175506607795, −5.67595031944522013313221045089, −5.39030718820709127941209345339, −5.33379504728974060162794953732, −4.92466216797136882337836470725, −4.48014821404409193064830801712, −4.34645696729541782834819138731, −4.21642320315638735892081658755, −4.05262928251473097624945113995, −3.54161578949348247210801353318, −3.02795612468554015068081966573, −2.95177043823742787400304234771, −2.53338165698185548864165943671, −2.40159277881274627052346039447, −1.57834413627581354642474715405, −1.03888422211361464049850295982, −0.60852986743194229708444574820, 0.60852986743194229708444574820, 1.03888422211361464049850295982, 1.57834413627581354642474715405, 2.40159277881274627052346039447, 2.53338165698185548864165943671, 2.95177043823742787400304234771, 3.02795612468554015068081966573, 3.54161578949348247210801353318, 4.05262928251473097624945113995, 4.21642320315638735892081658755, 4.34645696729541782834819138731, 4.48014821404409193064830801712, 4.92466216797136882337836470725, 5.33379504728974060162794953732, 5.39030718820709127941209345339, 5.67595031944522013313221045089, 5.94629681735106961175506607795, 6.13384615958499342549676490383, 6.49472017848852510188881082873, 6.76894716193023402380030785973, 6.81889230570599300259343450750, 7.00563683171543417120531968409, 7.37293913524197533799493645969, 7.50887323311184825290869264057, 7.62731377401260901881907374390

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