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Properties

Label 2-5070-13.12-c1-0-16
Degree $2$
Conductor $5070$
Sign $-0.277 - 0.960i$
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 + 2.32i·7-s + i·8-s + 9-s + 10-s + 5.34i·11-s − 12-s + 2.32·14-s + i·15-s + 16-s − 4·17-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 + 0.877i·7-s + 0.353i·8-s + 0.333·9-s + 0.316·10-s + 1.61i·11-s − 0.288·12-s + 0.620·14-s + 0.258i·15-s + 0.250·16-s − 0.970·17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $-0.277 - 0.960i$
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: \((2,\ 5070,\ (\ :1/2),\ -0.277 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.509368115\)
\(L(\frac12)\) \(\approx\) \(1.509368115\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.32iT - 7T^{2} \)
11 \( 1 - 5.34iT - 11T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 - 4.02iT - 19T^{2} \)
23 \( 1 - 4.93T + 23T^{2} \)
29 \( 1 - 4.29T + 29T^{2} \)
31 \( 1 + 3.47iT - 31T^{2} \)
37 \( 1 + 3.14iT - 37T^{2} \)
41 \( 1 - 2.64iT - 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 - 1.81iT - 47T^{2} \)
53 \( 1 + 5.48T + 53T^{2} \)
59 \( 1 + 6.78iT - 59T^{2} \)
61 \( 1 - 0.535T + 61T^{2} \)
67 \( 1 - 4.10iT - 67T^{2} \)
71 \( 1 + 15.8iT - 71T^{2} \)
73 \( 1 - 13.5iT - 73T^{2} \)
79 \( 1 + 7.96T + 79T^{2} \)
83 \( 1 - 11.3iT - 83T^{2} \)
89 \( 1 + 1.73iT - 89T^{2} \)
97 \( 1 - 16.1iT - 97T^{2} \)
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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

−8.497552308577330498181189444265, −7.910611712931542387394504490024, −7.00680222963340790115657361322, −6.42702178749939819614522633436, −5.31204711057870538496132150363, −4.62728885418793321422579908042, −3.86321679257701913752825198356, −2.88820239595938649400953516337, −2.26883023303013534321912409937, −1.55786568885365281837953535836, 0.37250810721874525667233295535, 1.33110579989696956611831560883, 2.84246023424958601074620047466, 3.50254304924878717019215611428, 4.45705696919788211834594909398, 4.98604685874216049340596567567, 5.95021402519829035777172750618, 6.79396654827697946712559959248, 7.16509587386889087555933219608, 8.212854602078789898452723127525

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