Properties

Label 8-567e4-1.1-c1e4-0-13
Degree $8$
Conductor $103355177121$
Sign $1$
Analytic cond. $420.185$
Root an. cond. $2.12779$
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  − 2·4-s + 10·7-s + 18·13-s + 4·16-s + 6·19-s − 8·25-s − 20·28-s − 6·31-s + 2·37-s + 2·43-s + 61·49-s − 36·52-s + 12·61-s − 16·64-s − 22·67-s + 6·73-s − 12·76-s − 10·79-s + 180·91-s − 36·97-s + 16·100-s + 2·109-s + 40·112-s + 40·121-s + 12·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 4-s + 3.77·7-s + 4.99·13-s + 16-s + 1.37·19-s − 8/5·25-s − 3.77·28-s − 1.07·31-s + 0.328·37-s + 0.304·43-s + 61/7·49-s − 4.99·52-s + 1.53·61-s − 2·64-s − 2.68·67-s + 0.702·73-s − 1.37·76-s − 1.12·79-s + 18.8·91-s − 3.65·97-s + 8/5·100-s + 0.191·109-s + 3.77·112-s + 3.63·121-s + 1.07·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{16} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(420.185\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{567} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{16} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.915783646\)
\(L(\frac12)\) \(\approx\) \(5.915783646\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
good2$C_2$$\times$$C_2^2$ \( ( 1 + p T^{2} )^{2}( 1 - p T^{2} + p^{2} T^{4} ) \)
5$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^3$ \( 1 - 10 T^{2} - 189 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^3$ \( 1 + 50 T^{2} + 1659 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^3$ \( 1 - 28 T^{2} - 897 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^2$ \( ( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 56 T^{2} + 927 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 98 T^{2} + 6795 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^3$ \( 1 - 94 T^{2} + 5355 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 6 T + 73 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
71$C_2^2$ \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
79$C_2^2$ \( ( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 112 T^{2} + 5655 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^3$ \( 1 - 154 T^{2} + 15795 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 + 18 T + 205 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.893871690998630528431346819007, −7.48422665484770251711252216365, −7.43363979037443924273980427361, −7.27843878491923342367940872344, −6.98575851100083715731601629340, −6.26631296290212038167848212272, −6.06503562516872818945650245285, −6.01629935633614839713288584799, −5.83488728820976416365292429234, −5.57437200410905588881297469539, −5.23656119156737199851146161712, −5.04911992710713684753711530551, −4.87631840434462117101426677532, −4.27078655163865553323861757996, −4.24719456645803767679318495207, −3.95650409302015532512979972719, −3.90868214594542927641564063072, −3.39229902308404054209432818102, −3.28164616695823343451397348105, −2.69063004478973744775882442865, −2.03010064006059237676133456416, −1.71545602195706601283614000430, −1.45745426210340890765996525544, −1.12378064174167236051209178033, −1.02025702418944601500794049073, 1.02025702418944601500794049073, 1.12378064174167236051209178033, 1.45745426210340890765996525544, 1.71545602195706601283614000430, 2.03010064006059237676133456416, 2.69063004478973744775882442865, 3.28164616695823343451397348105, 3.39229902308404054209432818102, 3.90868214594542927641564063072, 3.95650409302015532512979972719, 4.24719456645803767679318495207, 4.27078655163865553323861757996, 4.87631840434462117101426677532, 5.04911992710713684753711530551, 5.23656119156737199851146161712, 5.57437200410905588881297469539, 5.83488728820976416365292429234, 6.01629935633614839713288584799, 6.06503562516872818945650245285, 6.26631296290212038167848212272, 6.98575851100083715731601629340, 7.27843878491923342367940872344, 7.43363979037443924273980427361, 7.48422665484770251711252216365, 7.893871690998630528431346819007

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