Properties

Degree $2$
Conductor $82810$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 8-s − 3·9-s + 10-s − 4·11-s + 16-s − 2·17-s + 3·18-s − 20-s + 4·22-s + 25-s + 6·29-s + 8·31-s − 32-s + 2·34-s − 3·36-s + 10·37-s + 40-s + 2·41-s + 4·43-s − 4·44-s + 3·45-s + 8·47-s − 50-s − 2·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s − 9-s + 0.316·10-s − 1.20·11-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.223·20-s + 0.852·22-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s − 1/2·36-s + 1.64·37-s + 0.158·40-s + 0.312·41-s + 0.609·43-s − 0.603·44-s + 0.447·45-s + 1.16·47-s − 0.141·50-s − 0.274·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(82810\)    =    \(2 \cdot 5 \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{82810} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 82810,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.314440762\)
\(L(\frac12)\) \(\approx\) \(1.314440762\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
13 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.97115646822781, −13.52755892355645, −12.84040823442177, −12.43097369158631, −11.87473729669809, −11.30256943741797, −11.03175932003916, −10.48112168517816, −9.969783730527134, −9.439103320997674, −8.804309984012621, −8.377009155098884, −7.860466321372621, −7.682520079321848, −6.724808444098479, −6.399580400832965, −5.736229750463316, −5.156226251616683, −4.583462878885602, −3.893835337949077, −3.072736512249953, −2.534538506626088, −2.262389088156329, −0.9084468008594947, −0.5338441334780999, 0.5338441334780999, 0.9084468008594947, 2.262389088156329, 2.534538506626088, 3.072736512249953, 3.893835337949077, 4.583462878885602, 5.156226251616683, 5.736229750463316, 6.399580400832965, 6.724808444098479, 7.682520079321848, 7.860466321372621, 8.377009155098884, 8.804309984012621, 9.439103320997674, 9.969783730527134, 10.48112168517816, 11.03175932003916, 11.30256943741797, 11.87473729669809, 12.43097369158631, 12.84040823442177, 13.52755892355645, 13.97115646822781

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