Properties

Label 2-5290-1.1-c1-0-99
Degree $2$
Conductor $5290$
Sign $1$
Analytic cond. $42.2408$
Root an. cond. $6.49929$
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  − 2-s + 3.30·3-s + 4-s − 5-s − 3.30·6-s + 0.302·7-s − 8-s + 7.90·9-s + 10-s + 5.30·11-s + 3.30·12-s − 0.302·13-s − 0.302·14-s − 3.30·15-s + 16-s + 3.90·17-s − 7.90·18-s + 4.90·19-s − 20-s + 1.00·21-s − 5.30·22-s − 3.30·24-s + 25-s + 0.302·26-s + 16.2·27-s + 0.302·28-s + 4.60·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.90·3-s + 0.5·4-s − 0.447·5-s − 1.34·6-s + 0.114·7-s − 0.353·8-s + 2.63·9-s + 0.316·10-s + 1.59·11-s + 0.953·12-s − 0.0839·13-s − 0.0809·14-s − 0.852·15-s + 0.250·16-s + 0.947·17-s − 1.86·18-s + 1.12·19-s − 0.223·20-s + 0.218·21-s − 1.13·22-s − 0.674·24-s + 0.200·25-s + 0.0593·26-s + 3.11·27-s + 0.0572·28-s + 0.855·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5290\)    =    \(2 \cdot 5 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(42.2408\)
Root analytic conductor: \(6.49929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5290,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.505854253\)
\(L(\frac12)\) \(\approx\) \(3.505854253\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + T \)
23 \( 1 \)
good3 \( 1 - 3.30T + 3T^{2} \)
7 \( 1 - 0.302T + 7T^{2} \)
11 \( 1 - 5.30T + 11T^{2} \)
13 \( 1 + 0.302T + 13T^{2} \)
17 \( 1 - 3.90T + 17T^{2} \)
19 \( 1 - 4.90T + 19T^{2} \)
29 \( 1 - 4.60T + 29T^{2} \)
31 \( 1 - 2.90T + 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + 9.90T + 41T^{2} \)
43 \( 1 + 5.21T + 43T^{2} \)
47 \( 1 - 4.60T + 47T^{2} \)
53 \( 1 + 3.21T + 53T^{2} \)
59 \( 1 + 10.6T + 59T^{2} \)
61 \( 1 - 6.51T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 - 15.8T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 - 3.21T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 2.69T + 97T^{2} \)
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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

−8.248573402024787708120180275294, −7.76724255537707758042311995469, −6.98661838218681157121995456683, −6.56884801584253344478848150125, −5.13026902404799126415466140273, −4.14976797084711345520746040983, −3.39399312264718667133158518062, −2.99727562464875189347740904144, −1.73148530922489940735699376701, −1.17774502232057396099768266948, 1.17774502232057396099768266948, 1.73148530922489940735699376701, 2.99727562464875189347740904144, 3.39399312264718667133158518062, 4.14976797084711345520746040983, 5.13026902404799126415466140273, 6.56884801584253344478848150125, 6.98661838218681157121995456683, 7.76724255537707758042311995469, 8.248573402024787708120180275294

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