Properties

Label 2-5290-1.1-c1-0-136
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 + 2.68·3-s + 4-s + 5-s + 2.68·6-s + 4.59·7-s + 8-s + 4.22·9-s + 10-s − 5.13·11-s + 2.68·12-s − 1.22·13-s + 4.59·14-s + 2.68·15-s + 16-s + 4.68·17-s + 4.22·18-s + 4.59·19-s + 20-s + 12.3·21-s − 5.13·22-s + 2.68·24-s + 25-s − 1.22·26-s + 3.28·27-s + 4.59·28-s + 3.37·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.55·3-s + 0.5·4-s + 0.447·5-s + 1.09·6-s + 1.73·7-s + 0.353·8-s + 1.40·9-s + 0.316·10-s − 1.54·11-s + 0.775·12-s − 0.338·13-s + 1.22·14-s + 0.693·15-s + 0.250·16-s + 1.13·17-s + 0.995·18-s + 1.05·19-s + 0.223·20-s + 2.69·21-s − 1.09·22-s + 0.548·24-s + 0.200·25-s − 0.239·26-s + 0.632·27-s + 0.868·28-s + 0.626·29-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

\(L(1)\) \(\approx\) \(7.192384603\)
\(L(\frac12)\) \(\approx\) \(7.192384603\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.68T + 3T^{2} \)
7 \( 1 - 4.59T + 7T^{2} \)
11 \( 1 + 5.13T + 11T^{2} \)
13 \( 1 + 1.22T + 13T^{2} \)
17 \( 1 - 4.68T + 17T^{2} \)
19 \( 1 - 4.59T + 19T^{2} \)
29 \( 1 - 3.37T + 29T^{2} \)
31 \( 1 + 0.777T + 31T^{2} \)
37 \( 1 + 5.81T + 37T^{2} \)
41 \( 1 + 8.50T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 6.44T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 9.37T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + 15.6T + 67T^{2} \)
71 \( 1 - 1.31T + 71T^{2} \)
73 \( 1 + 4.44T + 73T^{2} \)
79 \( 1 - 4.88T + 79T^{2} \)
83 \( 1 - 3.81T + 83T^{2} \)
89 \( 1 + 8.93T + 89T^{2} \)
97 \( 1 - 18.0T + 97T^{2} \)
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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

−8.239858054728734994516883463912, −7.55201525237360954871242933708, −7.13109420931190164676078627216, −5.69566366078123926162477055323, −5.10332184670167621500059390160, −4.66610523362160838657701776569, −3.44751216237722378240481449055, −2.92297660399449295998224733165, −2.06754207227145485152371487459, −1.43852526136901948256193569870, 1.43852526136901948256193569870, 2.06754207227145485152371487459, 2.92297660399449295998224733165, 3.44751216237722378240481449055, 4.66610523362160838657701776569, 5.10332184670167621500059390160, 5.69566366078123926162477055323, 7.13109420931190164676078627216, 7.55201525237360954871242933708, 8.239858054728734994516883463912

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