Properties

Degree $2$
Conductor $250025$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 4·7-s + 3·8-s − 3·9-s + 4·11-s − 4·14-s − 16-s − 2·17-s + 3·18-s − 4·19-s − 4·22-s − 4·23-s − 4·28-s − 4·29-s − 2·31-s − 5·32-s + 2·34-s + 3·36-s − 6·37-s + 4·38-s + 6·41-s + 2·43-s − 4·44-s + 4·46-s − 6·47-s + 9·49-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.51·7-s + 1.06·8-s − 9-s + 1.20·11-s − 1.06·14-s − 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.917·19-s − 0.852·22-s − 0.834·23-s − 0.755·28-s − 0.742·29-s − 0.359·31-s − 0.883·32-s + 0.342·34-s + 1/2·36-s − 0.986·37-s + 0.648·38-s + 0.937·41-s + 0.304·43-s − 0.603·44-s + 0.589·46-s − 0.875·47-s + 9/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250025\)    =    \(5^{2} \cdot 73 \cdot 137\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{250025} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 250025,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
73 \( 1 + T \)
137 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.22171756125142, −12.39212662439572, −12.19529983649297, −11.45931575752390, −11.19690148422516, −10.85847623385638, −10.36096619463185, −9.672267260890868, −9.264056158267557, −8.812679197144059, −8.434413453004905, −8.161641670040483, −7.610494262284286, −7.089287774796574, −6.479276959530374, −5.908692772409275, −5.364740741862445, −4.904651996896597, −4.332698451706043, −3.931082224909676, −3.450814521075725, −2.329532052911555, −2.000697462362568, −1.435215584202858, −0.7324219264461374, 0, 0.7324219264461374, 1.435215584202858, 2.000697462362568, 2.329532052911555, 3.450814521075725, 3.931082224909676, 4.332698451706043, 4.904651996896597, 5.364740741862445, 5.908692772409275, 6.479276959530374, 7.089287774796574, 7.610494262284286, 8.161641670040483, 8.434413453004905, 8.812679197144059, 9.264056158267557, 9.672267260890868, 10.36096619463185, 10.85847623385638, 11.19690148422516, 11.45931575752390, 12.19529983649297, 12.39212662439572, 13.22171756125142

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