Properties

Label 2-91e2-1.1-c1-0-326
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  + 1.34·2-s + 2.05·3-s − 0.190·4-s + 3.56·5-s + 2.75·6-s − 2.94·8-s + 1.20·9-s + 4.79·10-s + 1.27·11-s − 0.390·12-s + 7.31·15-s − 3.58·16-s + 7.73·17-s + 1.61·18-s − 0.943·19-s − 0.679·20-s + 1.72·22-s + 1.64·23-s − 6.04·24-s + 7.72·25-s − 3.68·27-s + 4.04·29-s + 9.83·30-s − 5.15·31-s + 1.07·32-s + 2.62·33-s + 10.4·34-s + ⋯
L(s)  = 1  + 0.951·2-s + 1.18·3-s − 0.0951·4-s + 1.59·5-s + 1.12·6-s − 1.04·8-s + 0.400·9-s + 1.51·10-s + 0.385·11-s − 0.112·12-s + 1.88·15-s − 0.895·16-s + 1.87·17-s + 0.381·18-s − 0.216·19-s − 0.151·20-s + 0.366·22-s + 0.343·23-s − 1.23·24-s + 1.54·25-s − 0.709·27-s + 0.751·29-s + 1.79·30-s − 0.925·31-s + 0.189·32-s + 0.456·33-s + 1.78·34-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

\(L(1)\) \(\approx\) \(6.681779775\)
\(L(\frac12)\) \(\approx\) \(6.681779775\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.34T + 2T^{2} \)
3 \( 1 - 2.05T + 3T^{2} \)
5 \( 1 - 3.56T + 5T^{2} \)
11 \( 1 - 1.27T + 11T^{2} \)
17 \( 1 - 7.73T + 17T^{2} \)
19 \( 1 + 0.943T + 19T^{2} \)
23 \( 1 - 1.64T + 23T^{2} \)
29 \( 1 - 4.04T + 29T^{2} \)
31 \( 1 + 5.15T + 31T^{2} \)
37 \( 1 + 1.05T + 37T^{2} \)
41 \( 1 - 4.19T + 41T^{2} \)
43 \( 1 - 3.83T + 43T^{2} \)
47 \( 1 + 0.894T + 47T^{2} \)
53 \( 1 + 0.0799T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + 7.62T + 61T^{2} \)
67 \( 1 - 6.32T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 - 0.760T + 73T^{2} \)
79 \( 1 + 2.85T + 79T^{2} \)
83 \( 1 + 2.32T + 83T^{2} \)
89 \( 1 + 7.57T + 89T^{2} \)
97 \( 1 + 0.478T + 97T^{2} \)
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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

−7.898392051139300572848542460294, −6.98761965630546539344820186003, −6.18583648105993262150312022605, −5.58432838218665570056099680169, −5.17751101675477852199844881096, −4.12925106943044833982500937406, −3.41717630311052442223935855483, −2.81230397668993826074098960621, −2.11703605235426695684043166395, −1.10959023511523463277169636860, 1.10959023511523463277169636860, 2.11703605235426695684043166395, 2.81230397668993826074098960621, 3.41717630311052442223935855483, 4.12925106943044833982500937406, 5.17751101675477852199844881096, 5.58432838218665570056099680169, 6.18583648105993262150312022605, 6.98761965630546539344820186003, 7.898392051139300572848542460294

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