Properties

Label 2-5070-1.1-c1-0-25
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 − 2.60·7-s + 8-s + 9-s + 10-s − 12-s − 2.60·14-s − 15-s + 16-s + 2.60·17-s + 18-s + 2.60·19-s + 20-s + 2.60·21-s + 8.60·23-s − 24-s + 25-s − 27-s − 2.60·28-s − 2.60·29-s − 30-s + 6·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.984·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.288·12-s − 0.696·14-s − 0.258·15-s + 0.250·16-s + 0.631·17-s + 0.235·18-s + 0.597·19-s + 0.223·20-s + 0.568·21-s + 1.79·23-s − 0.204·24-s + 0.200·25-s − 0.192·27-s − 0.492·28-s − 0.483·29-s − 0.182·30-s + 1.07·31-s + 0.176·32-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

\(L(1)\) \(\approx\) \(2.569241998\)
\(L(\frac12)\) \(\approx\) \(2.569241998\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.60T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 - 2.60T + 17T^{2} \)
19 \( 1 - 2.60T + 19T^{2} \)
23 \( 1 - 8.60T + 23T^{2} \)
29 \( 1 + 2.60T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 5.21T + 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 5.21T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 5.21T + 59T^{2} \)
61 \( 1 - 3.21T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 + 5.21T + 71T^{2} \)
73 \( 1 - 8.60T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 - 17.2T + 83T^{2} \)
89 \( 1 + 0.788T + 89T^{2} \)
97 \( 1 - 8.60T + 97T^{2} \)
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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

−8.130850228679215748899614826416, −7.11961595923255558568530422163, −6.68207245460146034213515198336, −6.08098847275896407389133336575, −5.12690324195238066573339516480, −4.93114980408101210325267720506, −3.47270454043981370096301100649, −3.21829294345997772812634167558, −1.96699586581646016048602001970, −0.816459198591755542034641427459, 0.816459198591755542034641427459, 1.96699586581646016048602001970, 3.21829294345997772812634167558, 3.47270454043981370096301100649, 4.93114980408101210325267720506, 5.12690324195238066573339516480, 6.08098847275896407389133336575, 6.68207245460146034213515198336, 7.11961595923255558568530422163, 8.130850228679215748899614826416

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