Properties

Label 2-5070-13.12-c1-0-4
Degree $2$
Conductor $5070$
Sign $0.246 - 0.969i$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 3-s − 4-s i·5-s + i·6-s + 3.04i·7-s + i·8-s + 9-s − 10-s − 6.24i·11-s + 12-s + 3.04·14-s + i·15-s + 16-s − 2.69·17-s i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s + 1.15i·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s − 1.88i·11-s + 0.288·12-s + 0.814·14-s + 0.258i·15-s + 0.250·16-s − 0.652·17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $0.246 - 0.969i$
Analytic conductor: \(40.4841\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5070} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ 0.246 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4659119385\)
\(L(\frac12)\) \(\approx\) \(0.4659119385\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + T \)
5 \( 1 + iT \)
13 \( 1 \)
good7 \( 1 - 3.04iT - 7T^{2} \)
11 \( 1 + 6.24iT - 11T^{2} \)
17 \( 1 + 2.69T + 17T^{2} \)
19 \( 1 - 5.82iT - 19T^{2} \)
23 \( 1 - 5.62T + 23T^{2} \)
29 \( 1 + 5.14T + 29T^{2} \)
31 \( 1 - 3.53iT - 31T^{2} \)
37 \( 1 + 4.65iT - 37T^{2} \)
41 \( 1 - 3.77iT - 41T^{2} \)
43 \( 1 + 2.85T + 43T^{2} \)
47 \( 1 + 3.61iT - 47T^{2} \)
53 \( 1 - 0.664T + 53T^{2} \)
59 \( 1 + 12.9iT - 59T^{2} \)
61 \( 1 + 7.91T + 61T^{2} \)
67 \( 1 - 0.198iT - 67T^{2} \)
71 \( 1 - 0.374iT - 71T^{2} \)
73 \( 1 - 14.6iT - 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 + 6.83iT - 83T^{2} \)
89 \( 1 - 9.50iT - 89T^{2} \)
97 \( 1 - 0.335iT - 97T^{2} \)
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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

−8.583319393888445834594032848219, −7.957824928330762197731059875826, −6.78172099586468682897149040085, −5.82921030463739087310207151234, −5.60785214790020570245064123189, −4.79702321718466380241873742006, −3.73408724790913424148030118897, −3.08865171667360501394372639896, −2.02759929855120867056044307394, −1.03768067403991475902943316255, 0.15617528345317018793389955649, 1.47793356669188518486648387051, 2.67124140197049916935529837607, 3.90860469256281205758604388599, 4.59026340883535608557485779024, 4.98288025556001266193602737423, 6.11466367235236404941395932941, 6.84924550155221118165209126975, 7.29429530488744091308052184010, 7.54887138800022430151989687660

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