Properties

Degree $4$
Conductor $6502500$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s − 8·11-s + 16-s − 8·19-s + 20·29-s + 16·31-s + 36-s + 20·41-s + 8·44-s + 14·49-s − 24·59-s − 20·61-s − 64-s + 8·76-s + 16·79-s + 81-s + 12·89-s + 8·99-s − 20·101-s + 20·109-s − 20·116-s + 26·121-s − 16·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 2.41·11-s + 1/4·16-s − 1.83·19-s + 3.71·29-s + 2.87·31-s + 1/6·36-s + 3.12·41-s + 1.20·44-s + 2·49-s − 3.12·59-s − 2.56·61-s − 1/8·64-s + 0.917·76-s + 1.80·79-s + 1/9·81-s + 1.27·89-s + 0.804·99-s − 1.99·101-s + 1.91·109-s − 1.85·116-s + 2.36·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6502500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 17^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2550} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6502500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.579209925\)
\(L(\frac12)\) \(\approx\) \(1.579209925\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
17$C_2$ \( 1 + T^{2} \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
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$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−9.087910543834248305861046634847, −8.594596012387730146208750059529, −8.217657119251712969813260519890, −8.013495608938935115657463212527, −7.80323405699484606229033304585, −7.35693212743132773923222440700, −6.48525353000009579257296300101, −6.48349120520564199826544232607, −5.99743994823797004689189335285, −5.69750401876247945288874825377, −4.96271772592126370217239940426, −4.72689083643345171935497120211, −4.30483768612326852373799885486, −4.30393605622137117591546751804, −3.00107749095560770941295475832, −2.90052049639930777569910855694, −2.63209985225203183245732193592, −2.05669617851703378174457535933, −0.940197814882960083574972967081, −0.52462297082912301204589856869, 0.52462297082912301204589856869, 0.940197814882960083574972967081, 2.05669617851703378174457535933, 2.63209985225203183245732193592, 2.90052049639930777569910855694, 3.00107749095560770941295475832, 4.30393605622137117591546751804, 4.30483768612326852373799885486, 4.72689083643345171935497120211, 4.96271772592126370217239940426, 5.69750401876247945288874825377, 5.99743994823797004689189335285, 6.48349120520564199826544232607, 6.48525353000009579257296300101, 7.35693212743132773923222440700, 7.80323405699484606229033304585, 8.013495608938935115657463212527, 8.217657119251712969813260519890, 8.594596012387730146208750059529, 9.087910543834248305861046634847

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