Properties

Label 2-5070-13.12-c1-0-98
Degree $2$
Conductor $5070$
Sign $-0.832 - 0.554i$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 3-s − 4-s i·5-s i·6-s − 2i·7-s i·8-s + 9-s + 10-s − 4i·11-s + 12-s + 2·14-s + i·15-s + 16-s − 8·17-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 − 0.755i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s − 1.20i·11-s + 0.288·12-s + 0.534·14-s + 0.258i·15-s + 0.250·16-s − 1.94·17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $-0.832 - 0.554i$
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: \(1\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ -0.832 - 0.554i)\)

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 \( 1 - iT \)
3 \( 1 + T \)
5 \( 1 + iT \)
13 \( 1 \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
17 \( 1 + 8T + 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 2iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 8iT - 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 14iT - 89T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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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

−7.69524363933831705678297303833, −7.00385176977187641524415900809, −6.33117236701521112396022104275, −5.79640421327515474280345611260, −4.82102169894939032429475651892, −4.35644715598951209367070405300, −3.51802980583544664536812234277, −2.19284819031003011480493001001, −0.803750271251906704292505362934, 0, 1.90945856466375441765065330650, 2.08811721803671611353124900959, 3.36633959734645076296051979797, 4.30093706890926341643824962610, 4.77144264386310586897385757542, 5.91826796095453046104039957369, 6.23406995952728775956924325643, 7.28114190331287182662159880881, 7.890723838570164006515342968873

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