Properties

Degree $4$
Conductor $275625$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s − 9-s + 8·11-s + 5·16-s − 8·19-s + 4·29-s − 3·36-s + 4·41-s + 24·44-s − 49-s − 24·59-s − 4·61-s + 3·64-s − 24·76-s + 32·79-s + 81-s + 28·89-s − 8·99-s + 28·101-s + 36·109-s + 12·116-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 5·144-s + ⋯
L(s)  = 1  + 3/2·4-s − 1/3·9-s + 2.41·11-s + 5/4·16-s − 1.83·19-s + 0.742·29-s − 1/2·36-s + 0.624·41-s + 3.61·44-s − 1/7·49-s − 3.12·59-s − 0.512·61-s + 3/8·64-s − 2.75·76-s + 3.60·79-s + 1/9·81-s + 2.96·89-s − 0.804·99-s + 2.78·101-s + 3.44·109-s + 1.11·116-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.416·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(275625\)    =    \(3^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{525} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 275625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.92150\)
\(L(\frac12)\) \(\approx\) \(2.92150\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.95809906583212942849790701821, −10.80239565493072587102393066548, −10.42812947852719999368133858680, −9.756251620087751561418383755651, −9.140472566501612034019629535407, −9.005568505376896954722569084493, −8.488328608564253859675146718757, −7.73870928607150636665463250908, −7.54094628244471551744421440890, −6.78234151122684636589981585666, −6.34401412539296377470059259188, −6.31475191170401825389607281693, −5.92531618569511840937896131485, −4.65287061866001724993440637193, −4.61125913327929135093737781904, −3.49895542701816168695234282171, −3.45025563682784204589111900324, −2.29669120474052286253368683505, −1.96998235655500263440119339869, −1.08202254048993775485563266043, 1.08202254048993775485563266043, 1.96998235655500263440119339869, 2.29669120474052286253368683505, 3.45025563682784204589111900324, 3.49895542701816168695234282171, 4.61125913327929135093737781904, 4.65287061866001724993440637193, 5.92531618569511840937896131485, 6.31475191170401825389607281693, 6.34401412539296377470059259188, 6.78234151122684636589981585666, 7.54094628244471551744421440890, 7.73870928607150636665463250908, 8.488328608564253859675146718757, 9.005568505376896954722569084493, 9.140472566501612034019629535407, 9.756251620087751561418383755651, 10.42812947852719999368133858680, 10.80239565493072587102393066548, 10.95809906583212942849790701821

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