Properties

Label 2-91e2-1.1-c1-0-366
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  − 2.18·2-s + 1.75·3-s + 2.76·4-s − 2.11·5-s − 3.83·6-s − 1.67·8-s + 0.0920·9-s + 4.60·10-s + 5.76·11-s + 4.86·12-s − 3.71·15-s − 1.87·16-s + 1.64·17-s − 0.200·18-s + 2.67·19-s − 5.83·20-s − 12.5·22-s + 6.42·23-s − 2.94·24-s − 0.545·25-s − 5.11·27-s − 6.04·29-s + 8.10·30-s − 5.12·31-s + 7.45·32-s + 10.1·33-s − 3.58·34-s + ⋯
L(s)  = 1  − 1.54·2-s + 1.01·3-s + 1.38·4-s − 0.943·5-s − 1.56·6-s − 0.591·8-s + 0.0306·9-s + 1.45·10-s + 1.73·11-s + 1.40·12-s − 0.958·15-s − 0.469·16-s + 0.397·17-s − 0.0473·18-s + 0.612·19-s − 1.30·20-s − 2.68·22-s + 1.33·23-s − 0.600·24-s − 0.109·25-s − 0.984·27-s − 1.12·29-s + 1.47·30-s − 0.919·31-s + 1.31·32-s + 1.76·33-s − 0.614·34-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 \( 1 + 2.18T + 2T^{2} \)
3 \( 1 - 1.75T + 3T^{2} \)
5 \( 1 + 2.11T + 5T^{2} \)
11 \( 1 - 5.76T + 11T^{2} \)
17 \( 1 - 1.64T + 17T^{2} \)
19 \( 1 - 2.67T + 19T^{2} \)
23 \( 1 - 6.42T + 23T^{2} \)
29 \( 1 + 6.04T + 29T^{2} \)
31 \( 1 + 5.12T + 31T^{2} \)
37 \( 1 + 5.74T + 37T^{2} \)
41 \( 1 - 7.14T + 41T^{2} \)
43 \( 1 + 4.47T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 - 3.44T + 53T^{2} \)
59 \( 1 + 13.1T + 59T^{2} \)
61 \( 1 - 6.24T + 61T^{2} \)
67 \( 1 + 7.74T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 - 1.12T + 79T^{2} \)
83 \( 1 + 4.96T + 83T^{2} \)
89 \( 1 + 1.14T + 89T^{2} \)
97 \( 1 - 6.97T + 97T^{2} \)
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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

−7.65985569915888670099895127112, −7.18651021196107866971422755734, −6.56133468570494397757718772183, −5.49339860643013819444480724305, −4.36101440480386101698805551694, −3.59438862622294930673654970945, −3.08885952624629299592861918305, −1.87706057943070898978876178820, −1.22397620654511958981845069706, 0, 1.22397620654511958981845069706, 1.87706057943070898978876178820, 3.08885952624629299592861918305, 3.59438862622294930673654970945, 4.36101440480386101698805551694, 5.49339860643013819444480724305, 6.56133468570494397757718772183, 7.18651021196107866971422755734, 7.65985569915888670099895127112

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