Properties

Label 2-91e2-1.1-c1-0-181
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  + 0.180·2-s − 1.82·3-s − 1.96·4-s + 2.68·5-s − 0.330·6-s − 0.717·8-s + 0.334·9-s + 0.485·10-s + 2.69·11-s + 3.59·12-s − 4.90·15-s + 3.80·16-s + 4.76·17-s + 0.0604·18-s + 0.188·19-s − 5.28·20-s + 0.487·22-s + 4.39·23-s + 1.30·24-s + 2.21·25-s + 4.86·27-s + 7.08·29-s − 0.887·30-s − 3.69·31-s + 2.12·32-s − 4.91·33-s + 0.861·34-s + ⋯
L(s)  = 1  + 0.127·2-s − 1.05·3-s − 0.983·4-s + 1.20·5-s − 0.134·6-s − 0.253·8-s + 0.111·9-s + 0.153·10-s + 0.812·11-s + 1.03·12-s − 1.26·15-s + 0.951·16-s + 1.15·17-s + 0.0142·18-s + 0.0432·19-s − 1.18·20-s + 0.103·22-s + 0.917·23-s + 0.267·24-s + 0.443·25-s + 0.936·27-s + 1.31·29-s − 0.161·30-s − 0.664·31-s + 0.375·32-s − 0.856·33-s + 0.147·34-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

\(L(1)\) \(\approx\) \(1.633490927\)
\(L(\frac12)\) \(\approx\) \(1.633490927\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.180T + 2T^{2} \)
3 \( 1 + 1.82T + 3T^{2} \)
5 \( 1 - 2.68T + 5T^{2} \)
11 \( 1 - 2.69T + 11T^{2} \)
17 \( 1 - 4.76T + 17T^{2} \)
19 \( 1 - 0.188T + 19T^{2} \)
23 \( 1 - 4.39T + 23T^{2} \)
29 \( 1 - 7.08T + 29T^{2} \)
31 \( 1 + 3.69T + 31T^{2} \)
37 \( 1 - 7.95T + 37T^{2} \)
41 \( 1 - 5.42T + 41T^{2} \)
43 \( 1 + 8.01T + 43T^{2} \)
47 \( 1 + 1.84T + 47T^{2} \)
53 \( 1 + 7.07T + 53T^{2} \)
59 \( 1 - 7.58T + 59T^{2} \)
61 \( 1 + 0.411T + 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 + 3.34T + 71T^{2} \)
73 \( 1 + 14.2T + 73T^{2} \)
79 \( 1 - 9.11T + 79T^{2} \)
83 \( 1 - 16.5T + 83T^{2} \)
89 \( 1 + 5.89T + 89T^{2} \)
97 \( 1 + 0.451T + 97T^{2} \)
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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

−7.86697276005184321897179316526, −6.79318969731944958634148924967, −6.23218721650459775449769003545, −5.67377200848557309884705367661, −5.14780283231160462891860057154, −4.56337354354145316864222783629, −3.58037667315935237049238509548, −2.73309200139414944060959718864, −1.40802647404667995050420777135, −0.73821112401080174142257832791, 0.73821112401080174142257832791, 1.40802647404667995050420777135, 2.73309200139414944060959718864, 3.58037667315935237049238509548, 4.56337354354145316864222783629, 5.14780283231160462891860057154, 5.67377200848557309884705367661, 6.23218721650459775449769003545, 6.79318969731944958634148924967, 7.86697276005184321897179316526

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