Properties

Label 2-91e2-1.1-c1-0-108
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  − 1.76·2-s − 2.32·3-s + 1.12·4-s − 4.06·5-s + 4.11·6-s + 1.54·8-s + 2.41·9-s + 7.18·10-s − 2.90·11-s − 2.61·12-s + 9.46·15-s − 4.98·16-s − 3.37·17-s − 4.27·18-s + 1.18·19-s − 4.56·20-s + 5.13·22-s − 0.00746·23-s − 3.60·24-s + 11.5·25-s + 1.35·27-s − 8.00·29-s − 16.7·30-s + 0.679·31-s + 5.71·32-s + 6.76·33-s + 5.96·34-s + ⋯
L(s)  = 1  − 1.24·2-s − 1.34·3-s + 0.561·4-s − 1.81·5-s + 1.67·6-s + 0.547·8-s + 0.806·9-s + 2.27·10-s − 0.876·11-s − 0.754·12-s + 2.44·15-s − 1.24·16-s − 0.818·17-s − 1.00·18-s + 0.272·19-s − 1.02·20-s + 1.09·22-s − 0.00155·23-s − 0.736·24-s + 2.30·25-s + 0.259·27-s − 1.48·29-s − 3.05·30-s + 0.122·31-s + 1.00·32-s + 1.17·33-s + 1.02·34-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: \(1\)
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 \( 1 + 1.76T + 2T^{2} \)
3 \( 1 + 2.32T + 3T^{2} \)
5 \( 1 + 4.06T + 5T^{2} \)
11 \( 1 + 2.90T + 11T^{2} \)
17 \( 1 + 3.37T + 17T^{2} \)
19 \( 1 - 1.18T + 19T^{2} \)
23 \( 1 + 0.00746T + 23T^{2} \)
29 \( 1 + 8.00T + 29T^{2} \)
31 \( 1 - 0.679T + 31T^{2} \)
37 \( 1 + 2.58T + 37T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 - 4.33T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + 6.09T + 53T^{2} \)
59 \( 1 + 3.95T + 59T^{2} \)
61 \( 1 - 3.60T + 61T^{2} \)
67 \( 1 - 1.67T + 67T^{2} \)
71 \( 1 - 7.13T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + 1.82T + 83T^{2} \)
89 \( 1 - 3.60T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−7.50601946171800737533294363958, −7.03889872519656687642245502193, −6.35197073365614467706023962818, −5.20589052859451508645365410956, −4.80478533270673612253630918312, −4.02364150659001441802692788724, −3.11780405876963678914633156828, −1.76317515464497307783894594523, −0.55261232962706586016733319220, 0, 0.55261232962706586016733319220, 1.76317515464497307783894594523, 3.11780405876963678914633156828, 4.02364150659001441802692788724, 4.80478533270673612253630918312, 5.20589052859451508645365410956, 6.35197073365614467706023962818, 7.03889872519656687642245502193, 7.50601946171800737533294363958

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