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Properties

Label 2-780-13.4-c1-0-5
Degree $2$
Conductor $780$
Sign $0.999 + 0.00641i$
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  + (−0.5 + 0.866i)3-s + i·5-s + (3.40 − 1.96i)7-s + (−0.499 − 0.866i)9-s + (−1.14 − 0.658i)11-s + (1.86 − 3.08i)13-s + (−0.866 − 0.5i)15-s + (−0.276 − 0.478i)17-s + (4.69 − 2.71i)19-s + 3.93i·21-s + (−0.237 + 0.411i)23-s − 25-s + 0.999·27-s + (1.53 − 2.66i)29-s + 3.49i·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + 0.447i·5-s + (1.28 − 0.743i)7-s + (−0.166 − 0.288i)9-s + (−0.344 − 0.198i)11-s + (0.516 − 0.856i)13-s + (−0.223 − 0.129i)15-s + (−0.0670 − 0.116i)17-s + (1.07 − 0.622i)19-s + 0.858i·21-s + (−0.0495 + 0.0858i)23-s − 0.200·25-s + 0.192·27-s + (0.285 − 0.495i)29-s + 0.628i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.999 + 0.00641i$
Analytic conductor: \(6.22833\)
Root analytic conductor: \(2.49566\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{780} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 780,\ (\ :1/2),\ 0.999 + 0.00641i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61348 - 0.00517272i\)
\(L(\frac12)\) \(\approx\) \(1.61348 - 0.00517272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 - iT \)
13 \( 1 + (-1.86 + 3.08i)T \)
good7 \( 1 + (-3.40 + 1.96i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.14 + 0.658i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.276 + 0.478i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.69 + 2.71i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.237 - 0.411i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.53 + 2.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.49iT - 31T^{2} \)
37 \( 1 + (-0.407 - 0.235i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.53 - 1.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.697 - 1.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.71iT - 47T^{2} \)
53 \( 1 - 3.43T + 53T^{2} \)
59 \( 1 + (-11.2 + 6.50i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.313 - 0.542i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.0 - 6.37i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.51 - 2.60i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.95iT - 73T^{2} \)
79 \( 1 - 1.19T + 79T^{2} \)
83 \( 1 - 12.9iT - 83T^{2} \)
89 \( 1 + (15.5 + 8.95i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.54 - 2.62i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.42635097328173325851374641477, −9.663433750596522091275069565780, −8.456481253372474910654294417047, −7.78530091469389589088302812087, −6.90981253712620762411520616144, −5.67611441540901676884460480812, −4.93882228806845667846666608062, −3.93676742008873877095103571387, −2.79815252401242080118630960694, −1.04251642517399823611349929197, 1.33101826157860054755407470134, 2.31662382778158489029536465727, 4.01508771808170967067217494099, 5.13129767939934171866752854513, 5.68255459724457642140220698096, 6.87520840144286088220239329561, 7.85393695073455424438614726967, 8.481650889873278099594903056707, 9.288181550402974935154581043565, 10.40419230657191170376690145159

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