Properties

Label 2-29645-1.1-c1-0-2
Degree $2$
Conductor $29645$
Sign $1$
Analytic cond. $236.716$
Root an. cond. $15.3855$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 2·4-s + 5-s − 6·6-s + 6·9-s + 2·10-s − 6·12-s + 3·13-s − 3·15-s − 4·16-s − 3·17-s + 12·18-s + 6·19-s + 2·20-s − 4·23-s + 25-s + 6·26-s − 9·27-s + 29-s − 6·30-s − 6·31-s − 8·32-s − 6·34-s + 12·36-s + 12·38-s − 9·39-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.73·3-s + 4-s + 0.447·5-s − 2.44·6-s + 2·9-s + 0.632·10-s − 1.73·12-s + 0.832·13-s − 0.774·15-s − 16-s − 0.727·17-s + 2.82·18-s + 1.37·19-s + 0.447·20-s − 0.834·23-s + 1/5·25-s + 1.17·26-s − 1.73·27-s + 0.185·29-s − 1.09·30-s − 1.07·31-s − 1.41·32-s − 1.02·34-s + 2·36-s + 1.94·38-s − 1.44·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29645\)    =    \(5 \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(236.716\)
Root analytic conductor: \(15.3855\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{29645} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 29645,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.542978819\)
\(L(\frac12)\) \(\approx\) \(2.542978819\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 + p T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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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

−15.22276900476266, −14.49851681787317, −13.78847913674947, −13.68829429709990, −12.78909183062808, −12.66712983473821, −11.95892313705657, −11.59581107684449, −10.98672871697357, −10.74606205799362, −9.940307407013311, −9.315990438598312, −8.762022298782927, −7.608075738121515, −7.154623039171777, −6.434922449387035, −5.877404230798496, −5.792930130850741, −5.070305637111935, −4.523190770005430, −3.980614066621075, −3.278846610521838, −2.330685360250721, −1.466382548315798, −0.5550299924946579, 0.5550299924946579, 1.466382548315798, 2.330685360250721, 3.278846610521838, 3.980614066621075, 4.523190770005430, 5.070305637111935, 5.792930130850741, 5.877404230798496, 6.434922449387035, 7.154623039171777, 7.608075738121515, 8.762022298782927, 9.315990438598312, 9.940307407013311, 10.74606205799362, 10.98672871697357, 11.59581107684449, 11.95892313705657, 12.66712983473821, 12.78909183062808, 13.68829429709990, 13.78847913674947, 14.49851681787317, 15.22276900476266

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