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Properties

Label 2-780-12.11-c1-0-35
Degree $2$
Conductor $780$
Sign $-0.0105 - 0.999i$
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  + (1.38 + 0.261i)2-s + (−1.60 + 0.646i)3-s + (1.86 + 0.727i)4-s + i·5-s + (−2.40 + 0.478i)6-s + 1.32i·7-s + (2.39 + 1.49i)8-s + (2.16 − 2.07i)9-s + (−0.261 + 1.38i)10-s + 2.61·11-s + (−3.46 + 0.0366i)12-s + 13-s + (−0.346 + 1.83i)14-s + (−0.646 − 1.60i)15-s + (2.94 + 2.71i)16-s − 2.53i·17-s + ⋯
L(s)  = 1  + (0.982 + 0.185i)2-s + (−0.927 + 0.373i)3-s + (0.931 + 0.363i)4-s + 0.447i·5-s + (−0.980 + 0.195i)6-s + 0.499i·7-s + (0.848 + 0.529i)8-s + (0.720 − 0.692i)9-s + (−0.0827 + 0.439i)10-s + 0.788·11-s + (−0.999 + 0.0105i)12-s + 0.277·13-s + (−0.0924 + 0.491i)14-s + (−0.167 − 0.414i)15-s + (0.735 + 0.677i)16-s − 0.616i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0105 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0105 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.0105 - 0.999i$
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: \((2,\ 780,\ (\ :1/2),\ -0.0105 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54238 + 1.55880i\)
\(L(\frac12)\) \(\approx\) \(1.54238 + 1.55880i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 - 0.261i)T \)
3 \( 1 + (1.60 - 0.646i)T \)
5 \( 1 - iT \)
13 \( 1 - T \)
good7 \( 1 - 1.32iT - 7T^{2} \)
11 \( 1 - 2.61T + 11T^{2} \)
17 \( 1 + 2.53iT - 17T^{2} \)
19 \( 1 - 5.65iT - 19T^{2} \)
23 \( 1 + 9.33T + 23T^{2} \)
29 \( 1 - 4.17iT - 29T^{2} \)
31 \( 1 - 8.89iT - 31T^{2} \)
37 \( 1 - 5.76T + 37T^{2} \)
41 \( 1 + 0.238iT - 41T^{2} \)
43 \( 1 + 8.77iT - 43T^{2} \)
47 \( 1 - 1.43T + 47T^{2} \)
53 \( 1 + 7.39iT - 53T^{2} \)
59 \( 1 - 9.74T + 59T^{2} \)
61 \( 1 + 7.37T + 61T^{2} \)
67 \( 1 + 10.9iT - 67T^{2} \)
71 \( 1 + 4.35T + 71T^{2} \)
73 \( 1 + 2.82T + 73T^{2} \)
79 \( 1 + 7.31iT - 79T^{2} \)
83 \( 1 + 1.29T + 83T^{2} \)
89 \( 1 + 2.18iT - 89T^{2} \)
97 \( 1 - 13.6T + 97T^{2} \)
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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

−10.60978442036901145520782040692, −10.04284647004937062741553282500, −8.810624504639486174468477788799, −7.61343774791276929317717822590, −6.65752959349652777751572278918, −6.01632378246644509042322079858, −5.28700110856841799073833862682, −4.15585756833684923860986699258, −3.40290030936277731190639258945, −1.79899848589031932738246344658, 0.962480900950305706591038626531, 2.25203018094146445694493613820, 4.08604894007251606648446181538, 4.43421780595354804556457863452, 5.82399543263912043030475906118, 6.20733509845031767730980266334, 7.24339386443315202553495294523, 8.060713999178461475367529039246, 9.556031856575321969160193592213, 10.32810567316869751120972471791

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