Properties

Label 2-5070-1.1-c1-0-46
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 + 5.12·7-s − 8-s + 9-s − 10-s + 3.12·11-s + 12-s − 5.12·14-s + 15-s + 16-s − 2·17-s − 18-s − 6·19-s + 20-s + 5.12·21-s − 3.12·22-s − 3.12·23-s − 24-s + 25-s + 27-s + 5.12·28-s + 2·29-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 + 1.93·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.941·11-s + 0.288·12-s − 1.36·14-s + 0.258·15-s + 0.250·16-s − 0.485·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s + 1.11·21-s − 0.665·22-s − 0.651·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.968·28-s + 0.371·29-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: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.631777203\)
\(L(\frac12)\) \(\approx\) \(2.631777203\)
\(L(\frac{3}{2})\) 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 \( 1 - 5.12T + 7T^{2} \)
11 \( 1 - 3.12T + 11T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + 3.12T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 5.12T + 31T^{2} \)
37 \( 1 + 3.12T + 37T^{2} \)
41 \( 1 - 9.12T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 7.12T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 13.1T + 67T^{2} \)
71 \( 1 + 6.24T + 71T^{2} \)
73 \( 1 - 4.87T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 - 5.12T + 89T^{2} \)
97 \( 1 + 4.87T + 97T^{2} \)
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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

−8.295858778831265648647717397307, −7.76190986503516542735774339304, −6.96850114688887152038128490205, −6.22276615654099260342162773954, −5.32913397120662782787975404364, −4.41941654093257286246307742227, −3.84071302215742548291349741414, −2.30076780547346667120114416079, −1.98041498385356183753731884054, −1.02025477056700170093807526007, 1.02025477056700170093807526007, 1.98041498385356183753731884054, 2.30076780547346667120114416079, 3.84071302215742548291349741414, 4.41941654093257286246307742227, 5.32913397120662782787975404364, 6.22276615654099260342162773954, 6.96850114688887152038128490205, 7.76190986503516542735774339304, 8.295858778831265648647717397307

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