Properties

Label 2-5070-13.12-c1-0-53
Degree $2$
Conductor $5070$
Sign $0.277 + 0.960i$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 3-s − 4-s + i·5-s + i·6-s + 3i·7-s + i·8-s + 9-s + 10-s − 3.73i·11-s + 12-s + 3·14-s i·15-s + 16-s + 4·17-s i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s + 0.408i·6-s + 1.13i·7-s + 0.353i·8-s + 0.333·9-s + 0.316·10-s − 1.12i·11-s + 0.288·12-s + 0.801·14-s − 0.258i·15-s + 0.250·16-s + 0.970·17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $0.277 + 0.960i$
Analytic conductor: \(40.4841\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5070} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ 0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.176953377\)
\(L(\frac12)\) \(\approx\) \(1.176953377\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + T \)
5 \( 1 - iT \)
13 \( 1 \)
good7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 + 3.73iT - 11T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 - 2.26iT - 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 + 5.46T + 29T^{2} \)
31 \( 1 + 8.92iT - 31T^{2} \)
37 \( 1 + 7.92iT - 37T^{2} \)
41 \( 1 - 4iT - 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 - 0.464iT - 47T^{2} \)
53 \( 1 + 3.73T + 53T^{2} \)
59 \( 1 + 4.53iT - 59T^{2} \)
61 \( 1 + 7.46T + 61T^{2} \)
67 \( 1 - 5.46iT - 67T^{2} \)
71 \( 1 - 0.928iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 - 16.9T + 79T^{2} \)
83 \( 1 - 2.53iT - 83T^{2} \)
89 \( 1 + 10.1iT - 89T^{2} \)
97 \( 1 + 12.3iT - 97T^{2} \)
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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

−8.028981042405267479714550879774, −7.60212879516185304792838095941, −6.30638444635980929434413953698, −5.76272516776063713651467379375, −5.43605345475495981303458471354, −4.18439127943731529974593344757, −3.49164937411602068940364658509, −2.62505785345482465074887341911, −1.78187385341771302833441685523, −0.47188832565754415054088149750, 0.812840892031614584164812983219, 1.77182527361582038994531693629, 3.34844700392920738627505690887, 4.19198226656784861686529904621, 4.80080800642675224849020508396, 5.40757975314726165289853241647, 6.32082139997269966132276888563, 6.99838908128843550461697374530, 7.54645822131944421770604509491, 8.079866592957910359985386823152

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