Properties

Label 2-91e2-1.1-c1-0-368
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  + 2.58·2-s + 0.518·3-s + 4.70·4-s + 1.61·5-s + 1.34·6-s + 6.99·8-s − 2.73·9-s + 4.17·10-s + 2.70·11-s + 2.43·12-s + 0.835·15-s + 8.69·16-s + 3.12·17-s − 7.06·18-s + 3.68·19-s + 7.57·20-s + 7.00·22-s + 1.98·23-s + 3.62·24-s − 2.40·25-s − 2.97·27-s − 5.37·29-s + 2.16·30-s + 10.4·31-s + 8.52·32-s + 1.40·33-s + 8.09·34-s + ⋯
L(s)  = 1  + 1.83·2-s + 0.299·3-s + 2.35·4-s + 0.720·5-s + 0.547·6-s + 2.47·8-s − 0.910·9-s + 1.31·10-s + 0.815·11-s + 0.703·12-s + 0.215·15-s + 2.17·16-s + 0.758·17-s − 1.66·18-s + 0.844·19-s + 1.69·20-s + 1.49·22-s + 0.414·23-s + 0.739·24-s − 0.480·25-s − 0.571·27-s − 0.997·29-s + 0.395·30-s + 1.88·31-s + 1.50·32-s + 0.244·33-s + 1.38·34-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

\(L(1)\) \(\approx\) \(9.409688527\)
\(L(\frac12)\) \(\approx\) \(9.409688527\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.58T + 2T^{2} \)
3 \( 1 - 0.518T + 3T^{2} \)
5 \( 1 - 1.61T + 5T^{2} \)
11 \( 1 - 2.70T + 11T^{2} \)
17 \( 1 - 3.12T + 17T^{2} \)
19 \( 1 - 3.68T + 19T^{2} \)
23 \( 1 - 1.98T + 23T^{2} \)
29 \( 1 + 5.37T + 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 - 5.95T + 37T^{2} \)
41 \( 1 + 7.70T + 41T^{2} \)
43 \( 1 + 3.35T + 43T^{2} \)
47 \( 1 - 1.05T + 47T^{2} \)
53 \( 1 - 7.26T + 53T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 + 2.92T + 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 + 1.35T + 71T^{2} \)
73 \( 1 - 9.10T + 73T^{2} \)
79 \( 1 + 6.20T + 79T^{2} \)
83 \( 1 + 2.69T + 83T^{2} \)
89 \( 1 - 1.75T + 89T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−7.56110334326056537885317960857, −6.78825733705603323802904756219, −6.15936208539437881923672298292, −5.62645207876695223458942380755, −5.13362909306097286630776745013, −4.23866173547248953800756419370, −3.49759416327052507977567265823, −2.91765378558770906785325814412, −2.18627192837306150339072580661, −1.22919010345442936445694117404, 1.22919010345442936445694117404, 2.18627192837306150339072580661, 2.91765378558770906785325814412, 3.49759416327052507977567265823, 4.23866173547248953800756419370, 5.13362909306097286630776745013, 5.62645207876695223458942380755, 6.15936208539437881923672298292, 6.78825733705603323802904756219, 7.56110334326056537885317960857

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