Properties

Label 2-91e2-1.1-c1-0-464
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  + 2-s + 1.41·3-s − 4-s + 4.09·5-s + 1.41·6-s − 3·8-s − 0.999·9-s + 4.09·10-s − 3.79·11-s − 1.41·12-s + 5.79·15-s − 16-s − 1.26·17-s − 0.999·18-s − 2.82·19-s − 4.09·20-s − 3.79·22-s − 7.79·23-s − 4.24·24-s + 11.7·25-s − 5.65·27-s + 0.795·29-s + 5.79·30-s − 1.41·31-s + 5·32-s − 5.36·33-s − 1.26·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.816·3-s − 0.5·4-s + 1.83·5-s + 0.577·6-s − 1.06·8-s − 0.333·9-s + 1.29·10-s − 1.14·11-s − 0.408·12-s + 1.49·15-s − 0.250·16-s − 0.307·17-s − 0.235·18-s − 0.648·19-s − 0.916·20-s − 0.809·22-s − 1.62·23-s − 0.866·24-s + 2.35·25-s − 1.08·27-s + 0.147·29-s + 1.05·30-s − 0.254·31-s + 0.883·32-s − 0.934·33-s − 0.217·34-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 \( 1 - T + 2T^{2} \)
3 \( 1 - 1.41T + 3T^{2} \)
5 \( 1 - 4.09T + 5T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
17 \( 1 + 1.26T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 7.79T + 23T^{2} \)
29 \( 1 - 0.795T + 29T^{2} \)
31 \( 1 + 1.41T + 31T^{2} \)
37 \( 1 - 2.79T + 37T^{2} \)
41 \( 1 - 2.97T + 41T^{2} \)
43 \( 1 - 7.79T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 - 8.34T + 61T^{2} \)
67 \( 1 + 3.79T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 + 2.20T + 79T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 + 14.9T + 89T^{2} \)
97 \( 1 + 4.24T + 97T^{2} \)
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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

−7.58047990845871221093799371868, −6.39133762813086659769142639743, −5.90109119708965130033078066563, −5.49794841782714958544543515038, −4.67744596040200875337859874821, −3.93739922237364483028673655459, −2.77653583219558343960127506953, −2.58735428831980365958663128867, −1.66724417693147768874489687637, 0, 1.66724417693147768874489687637, 2.58735428831980365958663128867, 2.77653583219558343960127506953, 3.93739922237364483028673655459, 4.67744596040200875337859874821, 5.49794841782714958544543515038, 5.90109119708965130033078066563, 6.39133762813086659769142639743, 7.58047990845871221093799371868

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