Properties

Label 2-12635-1.1-c1-0-2
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s − 5-s + 7-s − 2·9-s − 3·11-s + 2·12-s − 5·13-s + 15-s + 4·16-s + 3·17-s + 2·20-s − 21-s − 6·23-s + 25-s + 5·27-s − 2·28-s − 3·29-s + 4·31-s + 3·33-s − 35-s + 4·36-s − 2·37-s + 5·39-s + 12·41-s − 10·43-s + 6·44-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.577·12-s − 1.38·13-s + 0.258·15-s + 16-s + 0.727·17-s + 0.447·20-s − 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.377·28-s − 0.557·29-s + 0.718·31-s + 0.522·33-s − 0.169·35-s + 2/3·36-s − 0.328·37-s + 0.800·39-s + 1.87·41-s − 1.52·43-s + 0.904·44-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: \(1\)
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 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−16.73833772606350, −16.07999318976877, −15.48879804047632, −14.59507147485527, −14.49886594548113, −13.83220515291904, −13.16116708915106, −12.36924537809527, −12.20903914769340, −11.53818690072543, −10.80472637385149, −10.24407977807533, −9.708730844030793, −9.075981835844921, −8.182802351905950, −7.937571190857328, −7.354492058876746, −6.303039086914539, −5.600891510660543, −5.105839934293347, −4.632497873715320, −3.817277033670284, −2.977970502058007, −2.148704702768943, −0.7527228293787584, 0, 0.7527228293787584, 2.148704702768943, 2.977970502058007, 3.817277033670284, 4.632497873715320, 5.105839934293347, 5.600891510660543, 6.303039086914539, 7.354492058876746, 7.937571190857328, 8.182802351905950, 9.075981835844921, 9.708730844030793, 10.24407977807533, 10.80472637385149, 11.53818690072543, 12.20903914769340, 12.36924537809527, 13.16116708915106, 13.83220515291904, 14.49886594548113, 14.59507147485527, 15.48879804047632, 16.07999318976877, 16.73833772606350

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