Properties

Label 2-5070-1.1-c1-0-93
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s + 1.04·7-s − 8-s + 9-s − 10-s − 1.35·11-s + 12-s − 1.04·14-s + 15-s + 16-s − 1.08·17-s − 18-s − 2.93·19-s + 20-s + 1.04·21-s + 1.35·22-s − 0.692·23-s − 24-s + 25-s + 27-s + 1.04·28-s − 2.37·29-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 + 0.396·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.409·11-s + 0.288·12-s − 0.280·14-s + 0.258·15-s + 0.250·16-s − 0.263·17-s − 0.235·18-s − 0.674·19-s + 0.223·20-s + 0.228·21-s + 0.289·22-s − 0.144·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.198·28-s − 0.441·29-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: \(1\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.04T + 7T^{2} \)
11 \( 1 + 1.35T + 11T^{2} \)
17 \( 1 + 1.08T + 17T^{2} \)
19 \( 1 + 2.93T + 19T^{2} \)
23 \( 1 + 0.692T + 23T^{2} \)
29 \( 1 + 2.37T + 29T^{2} \)
31 \( 1 + 9.85T + 31T^{2} \)
37 \( 1 + 9.26T + 37T^{2} \)
41 \( 1 + 2.84T + 41T^{2} \)
43 \( 1 + 4.45T + 43T^{2} \)
47 \( 1 - 3.31T + 47T^{2} \)
53 \( 1 - 0.664T + 53T^{2} \)
59 \( 1 - 1.96T + 59T^{2} \)
61 \( 1 - 3.24T + 61T^{2} \)
67 \( 1 + 6.91T + 67T^{2} \)
71 \( 1 + 2.29T + 71T^{2} \)
73 \( 1 + 3.36T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + 2.68T + 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + 1.56T + 97T^{2} \)
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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

−8.030388823059868280120344212138, −7.24320572168585258275483639072, −6.69849021219321335991623294129, −5.71483097439053974511835318230, −5.04636533501717605646594529686, −3.99288714035540905078431788199, −3.13006931986189083828914130790, −2.12292900676721814489047872300, −1.59158949651019088302826124550, 0, 1.59158949651019088302826124550, 2.12292900676721814489047872300, 3.13006931986189083828914130790, 3.99288714035540905078431788199, 5.04636533501717605646594529686, 5.71483097439053974511835318230, 6.69849021219321335991623294129, 7.24320572168585258275483639072, 8.030388823059868280120344212138

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