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Properties

Label 2-25383-1.1-c1-0-3
Degree $2$
Conductor $25383$
Sign $-1$
Analytic cond. $202.684$
Root an. cond. $14.2367$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $3$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s − 3·5-s + 2·6-s − 4·7-s + 9-s + 6·10-s − 4·11-s − 2·12-s − 6·13-s + 8·14-s + 3·15-s − 4·16-s − 6·17-s − 2·18-s − 4·19-s − 6·20-s + 4·21-s + 8·22-s − 6·23-s + 4·25-s + 12·26-s − 27-s − 8·28-s + 7·29-s − 6·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s − 1.34·5-s + 0.816·6-s − 1.51·7-s + 1/3·9-s + 1.89·10-s − 1.20·11-s − 0.577·12-s − 1.66·13-s + 2.13·14-s + 0.774·15-s − 16-s − 1.45·17-s − 0.471·18-s − 0.917·19-s − 1.34·20-s + 0.872·21-s + 1.70·22-s − 1.25·23-s + 4/5·25-s + 2.35·26-s − 0.192·27-s − 1.51·28-s + 1.29·29-s − 1.09·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25383 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25383 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25383\)    =    \(3 \cdot 8461\)
Sign: $-1$
Analytic conductor: \(202.684\)
Root analytic conductor: \(14.2367\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((2,\ 25383,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
8461 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−16.02734648968819, −15.93905533754971, −15.39904845566488, −15.00583106624744, −13.89843476304341, −13.30962538649480, −12.74051776117732, −12.24986183354338, −11.77191301423171, −11.19961414566392, −10.33667371689652, −10.23994040668202, −9.850590297684492, −8.801389746551619, −8.638776771329372, −7.815423153088893, −7.390416793823742, −6.791616407201063, −6.489813856359178, −5.458511114593659, −4.524924061678421, −4.341911274322121, −3.191975498781457, −2.581007430838877, −1.753661191209525, 0, 0, 0, 1.753661191209525, 2.581007430838877, 3.191975498781457, 4.341911274322121, 4.524924061678421, 5.458511114593659, 6.489813856359178, 6.791616407201063, 7.390416793823742, 7.815423153088893, 8.638776771329372, 8.801389746551619, 9.850590297684492, 10.23994040668202, 10.33667371689652, 11.19961414566392, 11.77191301423171, 12.24986183354338, 12.74051776117732, 13.30962538649480, 13.89843476304341, 15.00583106624744, 15.39904845566488, 15.93905533754971, 16.02734648968819

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