Properties

Label 2-12635-1.1-c1-0-8
Degree $2$
Conductor $12635$
Sign $1$
Analytic cond. $100.890$
Root an. cond. $10.0444$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 2·4-s + 5-s + 4·6-s + 7-s + 9-s − 2·10-s − 5·11-s − 4·12-s − 2·14-s − 2·15-s − 4·16-s − 4·17-s − 2·18-s + 2·20-s − 2·21-s + 10·22-s − 6·23-s + 25-s + 4·27-s + 2·28-s − 9·29-s + 4·30-s + 3·31-s + 8·32-s + 10·33-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 4-s + 0.447·5-s + 1.63·6-s + 0.377·7-s + 1/3·9-s − 0.632·10-s − 1.50·11-s − 1.15·12-s − 0.534·14-s − 0.516·15-s − 16-s − 0.970·17-s − 0.471·18-s + 0.447·20-s − 0.436·21-s + 2.13·22-s − 1.25·23-s + 1/5·25-s + 0.769·27-s + 0.377·28-s − 1.67·29-s + 0.730·30-s + 0.538·31-s + 1.41·32-s + 1.74·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12635 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12635 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12635\)    =    \(5 \cdot 7 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(100.890\)
Root analytic conductor: \(10.0444\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{12635} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 12635,\ (\ :1/2),\ 1)\)

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 \( 1 - T \)
19 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−16.85576256264699, −16.64082547792846, −16.00301911434067, −15.46737280160165, −14.86973392527509, −13.90397166387202, −13.35133468911745, −12.93230136667724, −11.99391399024821, −11.49590721158365, −10.94935086620012, −10.53533440707458, −10.01016222644479, −9.511590390814676, −8.569759762941403, −8.294082721116479, −7.495111308522073, −6.997539007015031, −6.111391461563249, −5.693178348736350, −4.876920138540062, −4.394271887450026, −2.987182264751275, −2.093104500657846, −1.458864592103876, 0, 0, 1.458864592103876, 2.093104500657846, 2.987182264751275, 4.394271887450026, 4.876920138540062, 5.693178348736350, 6.111391461563249, 6.997539007015031, 7.495111308522073, 8.294082721116479, 8.569759762941403, 9.511590390814676, 10.01016222644479, 10.53533440707458, 10.94935086620012, 11.49590721158365, 11.99391399024821, 12.93230136667724, 13.35133468911745, 13.90397166387202, 14.86973392527509, 15.46737280160165, 16.00301911434067, 16.64082547792846, 16.85576256264699

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