Properties

Label 2-5070-1.1-c1-0-87
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 + 4.24·7-s − 8-s + 9-s − 10-s − 1.91·11-s − 12-s − 4.24·14-s − 15-s + 16-s − 3.33·17-s − 18-s + 4.85·19-s + 20-s − 4.24·21-s + 1.91·22-s − 0.445·23-s + 24-s + 25-s − 27-s + 4.24·28-s − 8.56·29-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 + 1.60·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.576·11-s − 0.288·12-s − 1.13·14-s − 0.258·15-s + 0.250·16-s − 0.808·17-s − 0.235·18-s + 1.11·19-s + 0.223·20-s − 0.926·21-s + 0.407·22-s − 0.0927·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s + 0.802·28-s − 1.59·29-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 \( 1 - 4.24T + 7T^{2} \)
11 \( 1 + 1.91T + 11T^{2} \)
17 \( 1 + 3.33T + 17T^{2} \)
19 \( 1 - 4.85T + 19T^{2} \)
23 \( 1 + 0.445T + 23T^{2} \)
29 \( 1 + 8.56T + 29T^{2} \)
31 \( 1 + 5.29T + 31T^{2} \)
37 \( 1 + 1.75T + 37T^{2} \)
41 \( 1 + 3.24T + 41T^{2} \)
43 \( 1 + 1.97T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 + 1.14T + 53T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 - 1.13T + 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 - 2.14T + 71T^{2} \)
73 \( 1 + 5.15T + 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 - 9.49T + 83T^{2} \)
89 \( 1 - 1.25T + 89T^{2} \)
97 \( 1 - 5.01T + 97T^{2} \)
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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

−7.72931920499020830492441181188, −7.43478198496802426582989574338, −6.49931163443829656394400916073, −5.56309444106375095118007637567, −5.16487859340103535737479613759, −4.34585808968207905504375677561, −3.13113147031746869658640837313, −1.91062247819935596059076956954, −1.48552666373572429276451379960, 0, 1.48552666373572429276451379960, 1.91062247819935596059076956954, 3.13113147031746869658640837313, 4.34585808968207905504375677561, 5.16487859340103535737479613759, 5.56309444106375095118007637567, 6.49931163443829656394400916073, 7.43478198496802426582989574338, 7.72931920499020830492441181188

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