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Properties

Label 2-780-12.11-c1-0-40
Degree $2$
Conductor $780$
Sign $-0.642 - 0.766i$
Analytic cond. $6.22833$
Root an. cond. $2.49566$
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  + (1.13 + 0.844i)2-s + (0.684 + 1.59i)3-s + (0.573 + 1.91i)4-s − i·5-s + (−0.566 + 2.38i)6-s + 1.35i·7-s + (−0.968 + 2.65i)8-s + (−2.06 + 2.17i)9-s + (0.844 − 1.13i)10-s + 5.65·11-s + (−2.65 + 2.22i)12-s − 13-s + (−1.14 + 1.54i)14-s + (1.59 − 0.684i)15-s + (−3.34 + 2.19i)16-s + 4.77i·17-s + ⋯
L(s)  = 1  + (0.802 + 0.597i)2-s + (0.395 + 0.918i)3-s + (0.286 + 0.958i)4-s − 0.447i·5-s + (−0.231 + 0.972i)6-s + 0.513i·7-s + (−0.342 + 0.939i)8-s + (−0.687 + 0.726i)9-s + (0.267 − 0.358i)10-s + 1.70·11-s + (−0.766 + 0.642i)12-s − 0.277·13-s + (−0.306 + 0.412i)14-s + (0.410 − 0.176i)15-s + (−0.835 + 0.549i)16-s + 1.15i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(6.22833\)
Root analytic conductor: \(2.49566\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{780} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 780,\ (\ :1/2),\ -0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15625 + 2.47650i\)
\(L(\frac12)\) \(\approx\) \(1.15625 + 2.47650i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 - 0.844i)T \)
3 \( 1 + (-0.684 - 1.59i)T \)
5 \( 1 + iT \)
13 \( 1 + T \)
good7 \( 1 - 1.35iT - 7T^{2} \)
11 \( 1 - 5.65T + 11T^{2} \)
17 \( 1 - 4.77iT - 17T^{2} \)
19 \( 1 + 8.29iT - 19T^{2} \)
23 \( 1 + 3.30T + 23T^{2} \)
29 \( 1 + 3.90iT - 29T^{2} \)
31 \( 1 - 8.19iT - 31T^{2} \)
37 \( 1 - 3.51T + 37T^{2} \)
41 \( 1 - 1.39iT - 41T^{2} \)
43 \( 1 + 10.0iT - 43T^{2} \)
47 \( 1 + 6.45T + 47T^{2} \)
53 \( 1 - 4.68iT - 53T^{2} \)
59 \( 1 - 13.6T + 59T^{2} \)
61 \( 1 - 0.137T + 61T^{2} \)
67 \( 1 + 5.90iT - 67T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 + 3.48T + 73T^{2} \)
79 \( 1 + 12.9iT - 79T^{2} \)
83 \( 1 - 6.73T + 83T^{2} \)
89 \( 1 + 1.70iT - 89T^{2} \)
97 \( 1 - 1.74T + 97T^{2} \)
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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

−10.68711618382327379873288243045, −9.430202584497101194991929538717, −8.854029863807688901958677476112, −8.235331335879615098357696728353, −6.95606693538876471880870977642, −6.08966381026602130914157902810, −5.10032485979140406512849680596, −4.29249538928856672455616725073, −3.53033574475779166429626189656, −2.23020571462401614209020403271, 1.10809185948874019068740490621, 2.21902239137589094423544733509, 3.46903929150607609263282103501, 4.13964810381422813119912063599, 5.68949429527635437913898849968, 6.48780549154848123395039478833, 7.14215486538311207691363651581, 8.142236257826457526149262078859, 9.478610773481283124001654738374, 9.895792156184707047529072124209

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