Properties

Label 2-91e2-1.1-c1-0-45
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

\(L(1)\) \(\approx\) \(0.3387462925\)
\(L(\frac12)\) \(\approx\) \(0.3387462925\)
\(L(\frac{3}{2})\) 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

−8.034447467612236789404052839347, −7.15329696347352498831751914994, −6.41538798856379692381816640903, −5.49884558745809403468255347254, −4.98238568266355389695256548536, −4.55365210382172638964024375393, −3.50650203132451087035798192219, −3.00043797380912957165832739756, −1.25888486424146368621801481685, −0.35423015336388461633855987576, 0.35423015336388461633855987576, 1.25888486424146368621801481685, 3.00043797380912957165832739756, 3.50650203132451087035798192219, 4.55365210382172638964024375393, 4.98238568266355389695256548536, 5.49884558745809403468255347254, 6.41538798856379692381816640903, 7.15329696347352498831751914994, 8.034447467612236789404052839347

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