Properties

Label 2-47915-1.1-c1-0-2
Degree $2$
Conductor $47915$
Sign $-1$
Analytic cond. $382.603$
Root an. cond. $19.5602$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s − 7-s − 2·9-s − 2·10-s + 11-s + 2·12-s + 13-s − 2·14-s − 15-s − 4·16-s + 3·17-s − 4·18-s − 4·19-s − 2·20-s − 21-s + 2·22-s + 4·23-s + 25-s + 2·26-s − 5·27-s − 2·28-s + 3·29-s − 2·30-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s − 2/3·9-s − 0.632·10-s + 0.301·11-s + 0.577·12-s + 0.277·13-s − 0.534·14-s − 0.258·15-s − 16-s + 0.727·17-s − 0.942·18-s − 0.917·19-s − 0.447·20-s − 0.218·21-s + 0.426·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s − 0.962·27-s − 0.377·28-s + 0.557·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47915 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47915 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(47915\)    =    \(5 \cdot 7 \cdot 37^{2}\)
Sign: $-1$
Analytic conductor: \(382.603\)
Root analytic conductor: \(19.5602\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{47915} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 47915,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 + T \)
37 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 - T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−14.70579896069875, −14.31842844136945, −13.84837980251427, −13.30612635854217, −12.95017738883197, −12.28864907034558, −11.97142465963582, −11.41292451850247, −10.89816010522127, −10.27651850654239, −9.549919779126432, −8.914405272135311, −8.593379451319113, −7.963072689267273, −7.264174383835884, −6.656713562043143, −6.140315491400020, −5.639126128333161, −4.976242807556482, −4.360699244432827, −3.817560571891851, −3.261557652024564, −2.810758018905363, −2.207686478795953, −1.087696937401822, 0, 1.087696937401822, 2.207686478795953, 2.810758018905363, 3.261557652024564, 3.817560571891851, 4.360699244432827, 4.976242807556482, 5.639126128333161, 6.140315491400020, 6.656713562043143, 7.264174383835884, 7.963072689267273, 8.593379451319113, 8.914405272135311, 9.549919779126432, 10.27651850654239, 10.89816010522127, 11.41292451850247, 11.97142465963582, 12.28864907034558, 12.95017738883197, 13.30612635854217, 13.84837980251427, 14.31842844136945, 14.70579896069875

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