Properties

Label 8-72e4-1.1-c1e4-0-1
Degree $8$
Conductor $26873856$
Sign $1$
Analytic cond. $0.109254$
Root an. cond. $0.758236$
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  + 3-s + 5-s − 3·7-s + 3·9-s − 2·11-s − 5·13-s + 15-s − 10·17-s + 14·19-s − 3·21-s − 5·23-s + 2·25-s + 8·27-s + 3·29-s − 7·31-s − 2·33-s − 3·35-s + 12·37-s − 5·39-s + 12·41-s − 8·43-s + 3·45-s − 3·47-s + 8·49-s − 10·51-s − 20·53-s − 2·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.13·7-s + 9-s − 0.603·11-s − 1.38·13-s + 0.258·15-s − 2.42·17-s + 3.21·19-s − 0.654·21-s − 1.04·23-s + 2/5·25-s + 1.53·27-s + 0.557·29-s − 1.25·31-s − 0.348·33-s − 0.507·35-s + 1.97·37-s − 0.800·39-s + 1.87·41-s − 1.21·43-s + 0.447·45-s − 0.437·47-s + 8/7·49-s − 1.40·51-s − 2.74·53-s − 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(26873856\)    =    \(2^{12} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(0.109254\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 26873856,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7977493164\)
\(L(\frac12)\) \(\approx\) \(0.7977493164\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
good5$D_4\times C_2$ \( 1 - T - T^{2} + 8 T^{3} - 26 T^{4} + 8 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 + 3 T + T^{2} - 18 T^{3} - 48 T^{4} - 18 p T^{5} + p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 5 T + T^{2} - 10 T^{3} + 82 T^{4} - 10 p T^{5} + p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 5 T - 19 T^{2} - 10 T^{3} + 832 T^{4} - 10 p T^{5} - 19 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 18 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 + 7 T - 17 T^{2} + 28 T^{3} + 1876 T^{4} + 28 p T^{5} - 17 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 12 T + 59 T^{2} - 36 T^{3} - 360 T^{4} - 36 p T^{5} + 59 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 8 T - 5 T^{2} - 136 T^{3} + 160 T^{4} - 136 p T^{5} - 5 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 3 T - 79 T^{2} - 18 T^{3} + 5112 T^{4} - 18 p T^{5} - 79 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - T - 113 T^{2} + 8 T^{3} + 9214 T^{4} + 8 p T^{5} - 113 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 4 T - 89 T^{2} - 116 T^{3} + 5464 T^{4} - 116 p T^{5} - 89 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
73$D_{4}$ \( ( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 7 T - 113 T^{2} - 28 T^{3} + 16132 T^{4} - 28 p T^{5} - 113 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 + 25 T + 311 T^{2} + 3700 T^{3} + 39832 T^{4} + 3700 p T^{5} + 311 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
97$D_4\times C_2$ \( 1 - 8 T - 113 T^{2} + 136 T^{3} + 15712 T^{4} + 136 p T^{5} - 113 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−10.97484239703290360874038635021, −10.33316642614616915853722407192, −10.31658830517586373712031971696, −9.849785781643367104460999903642, −9.712103960320424840175102590290, −9.585166706487300286107305189198, −9.009342894345427883741841716162, −8.919556668285640819836691952917, −8.787589947031332386701770994099, −7.909028627599230400162437591922, −7.59561004699652374725667452262, −7.52659070916985522092147891192, −7.37605072270638361810659625308, −6.78796326320592240347188043600, −6.30787816442194039886959575366, −6.25523529252752935890943165888, −5.78935191560722170999123948828, −5.06172920739721406035955948384, −4.93524655292304400119499297670, −4.40959661098213397712541338428, −4.17387174021514465059944413867, −3.11732750306004008507032683652, −3.08366219427620883466983368697, −2.57422165224831863614353373906, −1.73332716400736499919266292061, 1.73332716400736499919266292061, 2.57422165224831863614353373906, 3.08366219427620883466983368697, 3.11732750306004008507032683652, 4.17387174021514465059944413867, 4.40959661098213397712541338428, 4.93524655292304400119499297670, 5.06172920739721406035955948384, 5.78935191560722170999123948828, 6.25523529252752935890943165888, 6.30787816442194039886959575366, 6.78796326320592240347188043600, 7.37605072270638361810659625308, 7.52659070916985522092147891192, 7.59561004699652374725667452262, 7.909028627599230400162437591922, 8.787589947031332386701770994099, 8.919556668285640819836691952917, 9.009342894345427883741841716162, 9.585166706487300286107305189198, 9.712103960320424840175102590290, 9.849785781643367104460999903642, 10.31658830517586373712031971696, 10.33316642614616915853722407192, 10.97484239703290360874038635021

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