Properties

Degree $2$
Conductor $116610$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s − 4·11-s + 12-s − 15-s + 16-s − 6·17-s − 18-s − 4·19-s − 20-s + 4·22-s − 23-s − 24-s + 25-s + 27-s − 2·29-s + 30-s − 32-s − 4·33-s + 6·34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.852·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.176·32-s − 0.696·33-s + 1.02·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116610\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 23\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{116610} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 116610,\ (\ :1/2),\ 1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 \)
23 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−14.04807729055553, −13.52863272760490, −13.01563811827790, −12.80507292923062, −12.10602233243385, −11.50604853326089, −11.07157133125891, −10.67583632882276, −10.07095188152199, −9.792408422958696, −8.974162935722501, −8.672695036196743, −8.179570805758699, −7.876930051158846, −7.160774863634649, −6.785337531078083, −6.273363259038292, −5.498757838304471, −4.893319686666648, −4.352940898325920, −3.723409810974249, −3.078920097029389, −2.425087358259283, −2.060865220293744, −1.274467227559031, 0, 0, 1.274467227559031, 2.060865220293744, 2.425087358259283, 3.078920097029389, 3.723409810974249, 4.352940898325920, 4.893319686666648, 5.498757838304471, 6.273363259038292, 6.785337531078083, 7.160774863634649, 7.876930051158846, 8.179570805758699, 8.672695036196743, 8.974162935722501, 9.792408422958696, 10.07095188152199, 10.67583632882276, 11.07157133125891, 11.50604853326089, 12.10602233243385, 12.80507292923062, 13.01563811827790, 13.52863272760490, 14.04807729055553

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