Properties

Label 2-5070-1.1-c1-0-91
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 + 3·7-s + 8-s + 9-s − 10-s − 0.267·11-s − 12-s + 3·14-s + 15-s + 16-s − 4·17-s + 18-s − 5.73·19-s − 20-s − 3·21-s − 0.267·22-s − 3.46·23-s − 24-s + 25-s − 27-s + 3·28-s + 1.46·29-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 + 1.13·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.0807·11-s − 0.288·12-s + 0.801·14-s + 0.258·15-s + 0.250·16-s − 0.970·17-s + 0.235·18-s − 1.31·19-s − 0.223·20-s − 0.654·21-s − 0.0571·22-s − 0.722·23-s − 0.204·24-s + 0.200·25-s − 0.192·27-s + 0.566·28-s + 0.271·29-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 \( 1 - 3T + 7T^{2} \)
11 \( 1 + 0.267T + 11T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + 5.73T + 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 1.46T + 29T^{2} \)
31 \( 1 + 4.92T + 31T^{2} \)
37 \( 1 - 5.92T + 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + 6.46T + 47T^{2} \)
53 \( 1 + 0.267T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 + 0.535T + 61T^{2} \)
67 \( 1 - 1.46T + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + 6.92T + 73T^{2} \)
79 \( 1 - 3.07T + 79T^{2} \)
83 \( 1 + 9.46T + 83T^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 + 8.39T + 97T^{2} \)
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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

−7.935552681287515422936024267515, −6.94710603968099400021566230738, −6.40580164335150059822681031794, −5.60360310655900281035659718470, −4.71569749720832739406751425049, −4.44179560312613737490121795082, −3.54140772495411128304689791254, −2.30628217005516636443956823120, −1.56333367285214444529380078693, 0, 1.56333367285214444529380078693, 2.30628217005516636443956823120, 3.54140772495411128304689791254, 4.44179560312613737490121795082, 4.71569749720832739406751425049, 5.60360310655900281035659718470, 6.40580164335150059822681031794, 6.94710603968099400021566230738, 7.935552681287515422936024267515

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