Properties

Label 2-91e2-1.1-c1-0-235
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.83·2-s + 0.853·3-s + 1.35·4-s − 2.62·5-s − 1.56·6-s + 1.17·8-s − 2.27·9-s + 4.81·10-s − 3.26·11-s + 1.15·12-s − 2.24·15-s − 4.87·16-s + 4.53·17-s + 4.16·18-s − 4.06·19-s − 3.56·20-s + 5.98·22-s − 4.53·23-s + 1.00·24-s + 1.89·25-s − 4.50·27-s − 1.42·29-s + 4.10·30-s + 2.80·31-s + 6.57·32-s − 2.78·33-s − 8.30·34-s + ⋯
L(s)  = 1  − 1.29·2-s + 0.492·3-s + 0.678·4-s − 1.17·5-s − 0.638·6-s + 0.416·8-s − 0.757·9-s + 1.52·10-s − 0.984·11-s + 0.334·12-s − 0.578·15-s − 1.21·16-s + 1.09·17-s + 0.980·18-s − 0.932·19-s − 0.797·20-s + 1.27·22-s − 0.945·23-s + 0.205·24-s + 0.378·25-s − 0.866·27-s − 0.264·29-s + 0.749·30-s + 0.503·31-s + 1.16·32-s − 0.485·33-s − 1.42·34-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: \(1\)
Selberg data: \((2,\ 8281,\ (\ :1/2),\ -1)\)

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)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + 1.83T + 2T^{2} \)
3 \( 1 - 0.853T + 3T^{2} \)
5 \( 1 + 2.62T + 5T^{2} \)
11 \( 1 + 3.26T + 11T^{2} \)
17 \( 1 - 4.53T + 17T^{2} \)
19 \( 1 + 4.06T + 19T^{2} \)
23 \( 1 + 4.53T + 23T^{2} \)
29 \( 1 + 1.42T + 29T^{2} \)
31 \( 1 - 2.80T + 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 - 2.84T + 41T^{2} \)
43 \( 1 - 9.72T + 43T^{2} \)
47 \( 1 - 9.44T + 47T^{2} \)
53 \( 1 - 5.26T + 53T^{2} \)
59 \( 1 - 2.56T + 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 - 1.98T + 67T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−7.73254381189945833887725626807, −7.37672324186099636333219703946, −6.18902831707994538644096776120, −5.50994594380492734336312718262, −4.39428737388495181359770947889, −3.90337791653529846972099510917, −2.82310046780077232960728487586, −2.22780521015848868957897687334, −0.861335467438693055237076129701, 0, 0.861335467438693055237076129701, 2.22780521015848868957897687334, 2.82310046780077232960728487586, 3.90337791653529846972099510917, 4.39428737388495181359770947889, 5.50994594380492734336312718262, 6.18902831707994538644096776120, 7.37672324186099636333219703946, 7.73254381189945833887725626807

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