Properties

Label 8-39e4-1.1-c3e4-0-1
Degree $8$
Conductor $2313441$
Sign $1$
Analytic cond. $28.0364$
Root an. cond. $1.51692$
Motivic weight $3$
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  − 40·7-s + 54·9-s − 128·16-s − 112·19-s + 616·31-s + 220·37-s + 800·49-s − 2.16e3·63-s + 1.76e3·67-s − 2.38e3·73-s + 2.18e3·81-s − 2.66e3·97-s + 1.29e3·109-s + 5.12e3·112-s + 127-s + 131-s + 4.48e3·133-s + 137-s + 139-s − 6.91e3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 506·169-s − 6.04e3·171-s + ⋯
L(s)  = 1  − 2.15·7-s + 2·9-s − 2·16-s − 1.35·19-s + 3.56·31-s + 0.977·37-s + 2.33·49-s − 4.31·63-s + 3.20·67-s − 3.81·73-s + 3·81-s − 2.78·97-s + 1.13·109-s + 4.31·112-s + 0.000698·127-s + 0.000666·131-s + 2.92·133-s + 0.000623·137-s + 0.000610·139-s − 4·144-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.230·169-s − 2.70·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(28.0364\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2313441,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.328173459\)
\(L(\frac12)\) \(\approx\) \(1.328173459\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 506 T^{2} + p^{6} T^{4} \)
good2$C_2$ \( ( 1 - p^{2} T + p^{3} T^{2} )^{2}( 1 + p^{2} T + p^{3} T^{2} )^{2} \)
5$C_2^2$ \( ( 1 + p^{6} T^{4} )^{2} \)
7$C_2$$\times$$C_2^2$ \( ( 1 + 20 T + p^{3} T^{2} )^{2}( 1 - 286 T^{2} + p^{6} T^{4} ) \)
11$C_2^2$ \( ( 1 + p^{6} T^{4} )^{2} \)
17$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
19$C_2$$\times$$C_2^2$ \( ( 1 + 56 T + p^{3} T^{2} )^{2}( 1 - 10582 T^{2} + p^{6} T^{4} ) \)
23$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
29$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
31$C_2$$\times$$C_2^2$ \( ( 1 - 308 T + p^{3} T^{2} )^{2}( 1 + 35282 T^{2} + p^{6} T^{4} ) \)
37$C_2$$\times$$C_2^2$ \( ( 1 - 110 T + p^{3} T^{2} )^{2}( 1 - 89206 T^{2} + p^{6} T^{4} ) \)
41$C_2^2$ \( ( 1 + p^{6} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 520 T + p^{3} T^{2} )^{2}( 1 + 520 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( ( 1 + p^{6} T^{4} )^{2} \)
53$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + p^{6} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 420838 T^{2} + p^{6} T^{4} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 - 880 T + p^{3} T^{2} )^{2}( 1 + 172874 T^{2} + p^{6} T^{4} ) \)
71$C_2^2$ \( ( 1 + p^{6} T^{4} )^{2} \)
73$C_2$$\times$$C_2^2$ \( ( 1 + 1190 T + p^{3} T^{2} )^{2}( 1 + 638066 T^{2} + p^{6} T^{4} ) \)
79$C_2^2$ \( ( 1 - 204622 T^{2} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + p^{6} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + p^{6} T^{4} )^{2} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + 1330 T + p^{3} T^{2} )^{2}( 1 - 56446 T^{2} + p^{6} T^{4} ) \)
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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

−11.67247590116403927743192449948, −11.47602965085814111631916940223, −10.95147971079917507375939622504, −10.46745627477810550433774939699, −10.38085660668039823711397929570, −10.01009997448948708431825629532, −9.606147852636555739952188230441, −9.444776700306060226029440777901, −9.309102329239299315269604686839, −8.601769481558970937853468140015, −8.155419048610548959214006919266, −8.141802739025896144043196910425, −7.13323445016229607942645609589, −6.96658573105494420842171104876, −6.82179053380510227777450328860, −6.42228761989683509583741006475, −6.12679362412166326822274238597, −5.55138954005615592418663929417, −4.49618854745559273726366511349, −4.38873976500028456531636509264, −4.29787692687587276082604923756, −3.28808317311945961300263030970, −2.74291168427077067100408965530, −2.05501370608218898058971466377, −0.66419974701740016513030352077, 0.66419974701740016513030352077, 2.05501370608218898058971466377, 2.74291168427077067100408965530, 3.28808317311945961300263030970, 4.29787692687587276082604923756, 4.38873976500028456531636509264, 4.49618854745559273726366511349, 5.55138954005615592418663929417, 6.12679362412166326822274238597, 6.42228761989683509583741006475, 6.82179053380510227777450328860, 6.96658573105494420842171104876, 7.13323445016229607942645609589, 8.141802739025896144043196910425, 8.155419048610548959214006919266, 8.601769481558970937853468140015, 9.309102329239299315269604686839, 9.444776700306060226029440777901, 9.606147852636555739952188230441, 10.01009997448948708431825629532, 10.38085660668039823711397929570, 10.46745627477810550433774939699, 10.95147971079917507375939622504, 11.47602965085814111631916940223, 11.67247590116403927743192449948

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