Properties

Label 2-5070-1.1-c1-0-1
Degree $2$
Conductor $5070$
Sign $1$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 5·7-s − 8-s + 9-s + 10-s − 3·11-s + 12-s + 5·14-s − 15-s + 16-s − 8·17-s − 18-s − 5·19-s − 20-s − 5·21-s + 3·22-s − 4·23-s − 24-s + 25-s + 27-s − 5·28-s − 4·29-s + 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s + 0.288·12-s + 1.33·14-s − 0.258·15-s + 1/4·16-s − 1.94·17-s − 0.235·18-s − 1.14·19-s − 0.223·20-s − 1.09·21-s + 0.639·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.944·28-s − 0.742·29-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: yes
Arithmetic: yes
Character: $\chi_{5070} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3157229217\)
\(L(\frac12)\) \(\approx\) \(0.3157229217\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−8.389753298913443470853712944656, −7.55164127065664012103482968139, −6.82677550398519109098126487887, −6.43171013588683796970282036283, −5.50131141718180918874804530132, −4.19575733074684163397955771306, −3.66840913133030569049909060586, −2.62344742699262395237953220949, −2.17170518156627208725673846946, −0.30097374533056560328987907482, 0.30097374533056560328987907482, 2.17170518156627208725673846946, 2.62344742699262395237953220949, 3.66840913133030569049909060586, 4.19575733074684163397955771306, 5.50131141718180918874804530132, 6.43171013588683796970282036283, 6.82677550398519109098126487887, 7.55164127065664012103482968139, 8.389753298913443470853712944656

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