Properties

Label 2-5070-1.1-c1-0-9
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 + 1.35·7-s − 8-s + 9-s + 10-s + 3.19·11-s − 12-s − 1.35·14-s + 15-s + 16-s − 2.04·17-s − 18-s − 5.34·19-s − 20-s − 1.35·21-s − 3.19·22-s − 8.32·23-s + 24-s + 25-s − 27-s + 1.35·28-s + 8.07·29-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 + 0.512·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.964·11-s − 0.288·12-s − 0.362·14-s + 0.258·15-s + 0.250·16-s − 0.496·17-s − 0.235·18-s − 1.22·19-s − 0.223·20-s − 0.296·21-s − 0.681·22-s − 1.73·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s + 0.256·28-s + 1.49·29-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

\(L(1)\) \(\approx\) \(0.9416862975\)
\(L(\frac12)\) \(\approx\) \(0.9416862975\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.35T + 7T^{2} \)
11 \( 1 - 3.19T + 11T^{2} \)
17 \( 1 + 2.04T + 17T^{2} \)
19 \( 1 + 5.34T + 19T^{2} \)
23 \( 1 + 8.32T + 23T^{2} \)
29 \( 1 - 8.07T + 29T^{2} \)
31 \( 1 + 0.185T + 31T^{2} \)
37 \( 1 + 2.46T + 37T^{2} \)
41 \( 1 - 12.2T + 41T^{2} \)
43 \( 1 + 2.91T + 43T^{2} \)
47 \( 1 - 13.0T + 47T^{2} \)
53 \( 1 - 5.74T + 53T^{2} \)
59 \( 1 + 1.62T + 59T^{2} \)
61 \( 1 - 5.28T + 61T^{2} \)
67 \( 1 - 1.55T + 67T^{2} \)
71 \( 1 + 2.32T + 71T^{2} \)
73 \( 1 + 5.11T + 73T^{2} \)
79 \( 1 + 1.42T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 - 3.37T + 89T^{2} \)
97 \( 1 + 4.74T + 97T^{2} \)
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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

−8.253833641425252630480857580540, −7.62835451793285502448403145515, −6.71307004264513881874196637460, −6.31917177135835036275046433639, −5.46451343059103464595566188802, −4.27401239050428083200440962019, −4.06629924642997500877030738959, −2.59255091548729693927150326837, −1.69617673626020252369096680466, −0.61614507003583976689337394429, 0.61614507003583976689337394429, 1.69617673626020252369096680466, 2.59255091548729693927150326837, 4.06629924642997500877030738959, 4.27401239050428083200440962019, 5.46451343059103464595566188802, 6.31917177135835036275046433639, 6.71307004264513881874196637460, 7.62835451793285502448403145515, 8.253833641425252630480857580540

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