Properties

Label 2-91e2-1.1-c1-0-109
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  − 0.680·2-s − 2.33·3-s − 1.53·4-s + 3.24·5-s + 1.58·6-s + 2.40·8-s + 2.44·9-s − 2.20·10-s − 5.33·11-s + 3.58·12-s − 7.57·15-s + 1.43·16-s + 2.66·17-s − 1.66·18-s + 0.696·19-s − 4.99·20-s + 3.62·22-s + 8.96·23-s − 5.61·24-s + 5.53·25-s + 1.28·27-s − 5.28·29-s + 5.15·30-s + 5.45·31-s − 5.79·32-s + 12.4·33-s − 1.81·34-s + ⋯
L(s)  = 1  − 0.480·2-s − 1.34·3-s − 0.768·4-s + 1.45·5-s + 0.648·6-s + 0.850·8-s + 0.816·9-s − 0.698·10-s − 1.60·11-s + 1.03·12-s − 1.95·15-s + 0.359·16-s + 0.647·17-s − 0.392·18-s + 0.159·19-s − 1.11·20-s + 0.773·22-s + 1.86·23-s − 1.14·24-s + 1.10·25-s + 0.247·27-s − 0.981·29-s + 0.940·30-s + 0.980·31-s − 1.02·32-s + 2.16·33-s − 0.311·34-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: \(0\)
Selberg data: \((2,\ 8281,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8171991238\)
\(L(\frac12)\) \(\approx\) \(0.8171991238\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.680T + 2T^{2} \)
3 \( 1 + 2.33T + 3T^{2} \)
5 \( 1 - 3.24T + 5T^{2} \)
11 \( 1 + 5.33T + 11T^{2} \)
17 \( 1 - 2.66T + 17T^{2} \)
19 \( 1 - 0.696T + 19T^{2} \)
23 \( 1 - 8.96T + 23T^{2} \)
29 \( 1 + 5.28T + 29T^{2} \)
31 \( 1 - 5.45T + 31T^{2} \)
37 \( 1 + 6.69T + 37T^{2} \)
41 \( 1 - 2.21T + 41T^{2} \)
43 \( 1 + 2.39T + 43T^{2} \)
47 \( 1 + 5.79T + 47T^{2} \)
53 \( 1 + 4.71T + 53T^{2} \)
59 \( 1 - 2.98T + 59T^{2} \)
61 \( 1 - 2.19T + 61T^{2} \)
67 \( 1 - 8.70T + 67T^{2} \)
71 \( 1 - 8.56T + 71T^{2} \)
73 \( 1 - 5.88T + 73T^{2} \)
79 \( 1 - 0.910T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 - 0.986T + 89T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−7.82249448328043399137461403539, −7.03201090950080935712417083307, −6.30191163683834353945289305097, −5.52499633722216320496708352657, −5.09038958527224858619547723294, −4.90948382526897738286084825534, −3.45950688085310795950050898193, −2.47475654179379871291795556303, −1.40795071410379295991725783962, −0.56000707070140132298633539503, 0.56000707070140132298633539503, 1.40795071410379295991725783962, 2.47475654179379871291795556303, 3.45950688085310795950050898193, 4.90948382526897738286084825534, 5.09038958527224858619547723294, 5.52499633722216320496708352657, 6.30191163683834353945289305097, 7.03201090950080935712417083307, 7.82249448328043399137461403539

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