Properties

Degree $8$
Conductor $15752961$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·4-s − 44·7-s + 64·16-s + 388·25-s − 704·28-s − 736·37-s − 760·43-s + 766·49-s − 1.02e3·64-s + 1.18e3·67-s + 3.34e3·79-s + 6.20e3·100-s − 3.37e3·109-s − 2.81e3·112-s + 4.84e3·121-s + 127-s + 131-s + 137-s + 139-s − 1.17e4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.01e3·169-s − 1.21e4·172-s + ⋯
L(s)  = 1  + 2·4-s − 2.37·7-s + 16-s + 3.10·25-s − 4.75·28-s − 3.27·37-s − 2.69·43-s + 2.23·49-s − 2·64-s + 2.15·67-s + 4.76·79-s + 6.20·100-s − 2.96·109-s − 2.37·112-s + 3.63·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 6.54·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.19·169-s − 5.39·172-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(15752961\)    =    \(3^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{63} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 15752961,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.17034\)
\(L(\frac12)\) \(\approx\) \(2.17034\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 + 22 T + p^{3} T^{2} )^{2} \)
good2$C_2^2$ \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 - 194 T^{2} + p^{6} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 20 p^{2} T^{2} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 3506 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 5830 T^{2} + p^{6} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 5726 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 18284 T^{2} + p^{6} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 32936 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 2750 T^{2} + p^{6} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 184 T + p^{3} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 126742 T^{2} + p^{6} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 190 T + p^{3} T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 205870 T^{2} + p^{6} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 169736 T^{2} + p^{6} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 403654 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 24170 T^{2} + p^{6} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 296 T + p^{3} T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 607244 T^{2} + p^{6} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 130682 T^{2} + p^{6} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 836 T + p^{3} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 350042 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 926422 T^{2} + p^{6} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 1504778 T^{2} + p^{6} T^{4} )^{2} \)
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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

−10.61877278634734939034060198100, −10.55939556092226575416043890539, −9.904028001666633793903900639830, −9.886499631655229577787961530760, −9.522816736914264142086769371306, −8.903497997729015406827512508557, −8.865301933401142896318093669180, −8.615087947156589652757809777505, −7.83296546528896013648912018195, −7.81414153030141916024143296395, −6.93601546209825179214630967095, −6.84292784910064500924589121149, −6.74280033067996113785743085337, −6.54871305090730183721877281609, −6.41624307458562817545313799155, −5.58651069511856230021107528563, −5.19318646703473447541040739388, −4.98468889737987714349882020274, −4.18225306365907447239200869850, −3.32880601330551707161350680948, −3.26560050153144205464053973087, −3.03008392180011171523769046824, −2.21729374725793309533415613459, −1.78433107398828365457187679253, −0.53856234751400809301860077153, 0.53856234751400809301860077153, 1.78433107398828365457187679253, 2.21729374725793309533415613459, 3.03008392180011171523769046824, 3.26560050153144205464053973087, 3.32880601330551707161350680948, 4.18225306365907447239200869850, 4.98468889737987714349882020274, 5.19318646703473447541040739388, 5.58651069511856230021107528563, 6.41624307458562817545313799155, 6.54871305090730183721877281609, 6.74280033067996113785743085337, 6.84292784910064500924589121149, 6.93601546209825179214630967095, 7.81414153030141916024143296395, 7.83296546528896013648912018195, 8.615087947156589652757809777505, 8.865301933401142896318093669180, 8.903497997729015406827512508557, 9.522816736914264142086769371306, 9.886499631655229577787961530760, 9.904028001666633793903900639830, 10.55939556092226575416043890539, 10.61877278634734939034060198100

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