Properties

Label 2-91e2-1.1-c1-0-183
Degree $2$
Conductor $8281$
Sign $1$
Analytic cond. $66.1241$
Root an. cond. $8.13167$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.30·2-s − 1.47·3-s + 3.30·4-s + 0.847·5-s − 3.39·6-s + 3.00·8-s − 0.829·9-s + 1.95·10-s − 1.50·11-s − 4.86·12-s − 1.24·15-s + 0.313·16-s + 2.07·17-s − 1.90·18-s − 0.0474·19-s + 2.80·20-s − 3.46·22-s − 7.81·23-s − 4.42·24-s − 4.28·25-s + 5.64·27-s + 1.35·29-s − 2.87·30-s + 7.86·31-s − 5.29·32-s + 2.21·33-s + 4.77·34-s + ⋯
L(s)  = 1  + 1.62·2-s − 0.850·3-s + 1.65·4-s + 0.378·5-s − 1.38·6-s + 1.06·8-s − 0.276·9-s + 0.617·10-s − 0.453·11-s − 1.40·12-s − 0.322·15-s + 0.0782·16-s + 0.502·17-s − 0.450·18-s − 0.0108·19-s + 0.626·20-s − 0.738·22-s − 1.63·23-s − 0.904·24-s − 0.856·25-s + 1.08·27-s + 0.252·29-s − 0.524·30-s + 1.41·31-s − 0.935·32-s + 0.385·33-s + 0.818·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8281\)    =    \(7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(66.1241\)
Root analytic conductor: \(8.13167\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8281} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8281,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.730391881\)
\(L(\frac12)\) \(\approx\) \(3.730391881\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 - 2.30T + 2T^{2} \)
3 \( 1 + 1.47T + 3T^{2} \)
5 \( 1 - 0.847T + 5T^{2} \)
11 \( 1 + 1.50T + 11T^{2} \)
17 \( 1 - 2.07T + 17T^{2} \)
19 \( 1 + 0.0474T + 19T^{2} \)
23 \( 1 + 7.81T + 23T^{2} \)
29 \( 1 - 1.35T + 29T^{2} \)
31 \( 1 - 7.86T + 31T^{2} \)
37 \( 1 - 6.70T + 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 - 9.26T + 43T^{2} \)
47 \( 1 - 0.360T + 47T^{2} \)
53 \( 1 - 2.71T + 53T^{2} \)
59 \( 1 - 1.64T + 59T^{2} \)
61 \( 1 - 4.52T + 61T^{2} \)
67 \( 1 + 2.04T + 67T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 - 6.76T + 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 + 17.5T + 89T^{2} \)
97 \( 1 - 0.426T + 97T^{2} \)
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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

−7.66819664457469855618551051775, −6.60415326542916891772315072866, −6.09083744146986008997318656129, −5.73623054117521124566251765064, −5.11752291666071020331518941224, −4.35815769363415312898334611851, −3.77442309083307995693511457488, −2.69681723534986551815302163906, −2.23237893311081306028734741547, −0.74853910896985506728218575008, 0.74853910896985506728218575008, 2.23237893311081306028734741547, 2.69681723534986551815302163906, 3.77442309083307995693511457488, 4.35815769363415312898334611851, 5.11752291666071020331518941224, 5.73623054117521124566251765064, 6.09083744146986008997318656129, 6.60415326542916891772315072866, 7.66819664457469855618551051775

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