Properties

Label 2-5070-1.1-c1-0-18
Degree $2$
Conductor $5070$
Sign $1$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
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-s + 4-s + 5-s − 6-s − 4.64·7-s − 8-s + 9-s − 10-s + 4.40·11-s + 12-s + 4.64·14-s + 15-s + 16-s + 4·17-s − 18-s − 8.04·19-s + 20-s − 4.64·21-s − 4.40·22-s − 0.976·23-s − 24-s + 25-s + 27-s − 4.64·28-s − 4.31·29-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.75·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.32·11-s + 0.288·12-s + 1.24·14-s + 0.258·15-s + 0.250·16-s + 0.970·17-s − 0.235·18-s − 1.84·19-s + 0.223·20-s − 1.01·21-s − 0.938·22-s − 0.203·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s − 0.877·28-s − 0.800·29-s − 0.182·30-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.495435495\)
\(L(\frac12)\) \(\approx\) \(1.495435495\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 \)
good7 \( 1 + 4.64T + 7T^{2} \)
11 \( 1 - 4.40T + 11T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 8.04T + 19T^{2} \)
23 \( 1 + 0.976T + 23T^{2} \)
29 \( 1 + 4.31T + 29T^{2} \)
31 \( 1 + 6.44T + 31T^{2} \)
37 \( 1 - 3.79T + 37T^{2} \)
41 \( 1 - 7.28T + 41T^{2} \)
43 \( 1 - 0.716T + 43T^{2} \)
47 \( 1 - 9.75T + 47T^{2} \)
53 \( 1 - 13.5T + 53T^{2} \)
59 \( 1 - 2.18T + 59T^{2} \)
61 \( 1 - 7.46T + 61T^{2} \)
67 \( 1 - 1.82T + 67T^{2} \)
71 \( 1 + 7.95T + 71T^{2} \)
73 \( 1 + 4.36T + 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 + 3.51T + 83T^{2} \)
89 \( 1 - 8.17T + 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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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

−8.534065084845908665823442191660, −7.34384379849874810034569498183, −6.99349073764660608857573279523, −6.05667157974529186756972720171, −5.84505304073339319997011884007, −4.12319295560417543655870025919, −3.68628622548320028319888550150, −2.71817608386031528643137313268, −1.93504206259727481857262766316, −0.70895536765787174477528688077, 0.70895536765787174477528688077, 1.93504206259727481857262766316, 2.71817608386031528643137313268, 3.68628622548320028319888550150, 4.12319295560417543655870025919, 5.84505304073339319997011884007, 6.05667157974529186756972720171, 6.99349073764660608857573279523, 7.34384379849874810034569498183, 8.534065084845908665823442191660

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