Properties

Label 2-91e2-1.1-c1-0-128
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  − 2.37·2-s − 0.696·3-s + 3.64·4-s − 2.62·5-s + 1.65·6-s − 3.90·8-s − 2.51·9-s + 6.23·10-s + 4.46·11-s − 2.53·12-s + 1.82·15-s + 1.98·16-s − 2.35·17-s + 5.97·18-s + 6.82·19-s − 9.56·20-s − 10.6·22-s + 1.91·23-s + 2.71·24-s + 1.89·25-s + 3.84·27-s + 7.24·29-s − 4.34·30-s + 7.55·31-s + 3.09·32-s − 3.11·33-s + 5.58·34-s + ⋯
L(s)  = 1  − 1.67·2-s − 0.402·3-s + 1.82·4-s − 1.17·5-s + 0.675·6-s − 1.37·8-s − 0.838·9-s + 1.97·10-s + 1.34·11-s − 0.732·12-s + 0.472·15-s + 0.495·16-s − 0.570·17-s + 1.40·18-s + 1.56·19-s − 2.13·20-s − 2.26·22-s + 0.398·23-s + 0.554·24-s + 0.378·25-s + 0.739·27-s + 1.34·29-s − 0.793·30-s + 1.35·31-s + 0.546·32-s − 0.541·33-s + 0.958·34-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8281,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5913676221\)
\(L(\frac12)\) \(\approx\) \(0.5913676221\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.37T + 2T^{2} \)
3 \( 1 + 0.696T + 3T^{2} \)
5 \( 1 + 2.62T + 5T^{2} \)
11 \( 1 - 4.46T + 11T^{2} \)
17 \( 1 + 2.35T + 17T^{2} \)
19 \( 1 - 6.82T + 19T^{2} \)
23 \( 1 - 1.91T + 23T^{2} \)
29 \( 1 - 7.24T + 29T^{2} \)
31 \( 1 - 7.55T + 31T^{2} \)
37 \( 1 + 5.94T + 37T^{2} \)
41 \( 1 - 0.271T + 41T^{2} \)
43 \( 1 + 3.28T + 43T^{2} \)
47 \( 1 - 8.10T + 47T^{2} \)
53 \( 1 - 7.19T + 53T^{2} \)
59 \( 1 - 9.04T + 59T^{2} \)
61 \( 1 - 7.88T + 61T^{2} \)
67 \( 1 + 7.00T + 67T^{2} \)
71 \( 1 - 6.84T + 71T^{2} \)
73 \( 1 + 0.348T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 - 13.4T + 83T^{2} \)
89 \( 1 + 6.00T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−8.029936093745014560081295491894, −7.12834964971220382641184542481, −6.85144987954259450936235569062, −6.06443091711285152776515957931, −5.07452585396902599534226522114, −4.18654030973354813736719584105, −3.31116978702665780990491662573, −2.48759510977355697433417615062, −1.16922290492667178425994318248, −0.60219919581535590650681902505, 0.60219919581535590650681902505, 1.16922290492667178425994318248, 2.48759510977355697433417615062, 3.31116978702665780990491662573, 4.18654030973354813736719584105, 5.07452585396902599534226522114, 6.06443091711285152776515957931, 6.85144987954259450936235569062, 7.12834964971220382641184542481, 8.029936093745014560081295491894

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