Properties

Label 2-91e2-1.1-c1-0-105
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  + 0.381·2-s − 2.23·3-s − 1.85·4-s − 2.23·5-s − 0.854·6-s − 1.47·8-s + 2.00·9-s − 0.854·10-s + 3·11-s + 4.14·12-s + 5.00·15-s + 3.14·16-s + 7.47·17-s + 0.763·18-s + 3·19-s + 4.14·20-s + 1.14·22-s − 3.76·23-s + 3.29·24-s + 2.23·27-s − 4.47·29-s + 1.90·30-s + 5·31-s + 4.14·32-s − 6.70·33-s + 2.85·34-s − 3.70·36-s + ⋯
L(s)  = 1  + 0.270·2-s − 1.29·3-s − 0.927·4-s − 0.999·5-s − 0.348·6-s − 0.520·8-s + 0.666·9-s − 0.270·10-s + 0.904·11-s + 1.19·12-s + 1.29·15-s + 0.786·16-s + 1.81·17-s + 0.180·18-s + 0.688·19-s + 0.927·20-s + 0.244·22-s − 0.784·23-s + 0.671·24-s + 0.430·27-s − 0.830·29-s + 0.348·30-s + 0.898·31-s + 0.732·32-s − 1.16·33-s + 0.489·34-s − 0.618·36-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

\(L(1)\) \(\approx\) \(0.8219213009\)
\(L(\frac12)\) \(\approx\) \(0.8219213009\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.381T + 2T^{2} \)
3 \( 1 + 2.23T + 3T^{2} \)
5 \( 1 + 2.23T + 5T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
17 \( 1 - 7.47T + 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 + 3.76T + 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 - 8.70T + 37T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 1.47T + 47T^{2} \)
53 \( 1 - 1.47T + 53T^{2} \)
59 \( 1 - 7.47T + 59T^{2} \)
61 \( 1 + 3T + 61T^{2} \)
67 \( 1 - 3T + 67T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 2.23T + 89T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−7.982820559881118595989684120727, −7.03618598381248319923111028360, −6.20328545158585072117424451383, −5.62405833519709966136408854330, −5.12242155758843402766723540319, −4.21939438976793778270733002014, −3.83523993381332403445586364991, −3.00538890566313823792091989657, −1.24658112869423572569226566663, −0.54379000206957633115181158789, 0.54379000206957633115181158789, 1.24658112869423572569226566663, 3.00538890566313823792091989657, 3.83523993381332403445586364991, 4.21939438976793778270733002014, 5.12242155758843402766723540319, 5.62405833519709966136408854330, 6.20328545158585072117424451383, 7.03618598381248319923111028360, 7.982820559881118595989684120727

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