↑

Properties

Label 2-780-13.10-c1-0-8
Degree $2$
Conductor $780$
Sign $-0.793 + 0.608i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + i·5-s + (−3.30 − 1.90i)7-s + (−0.499 + 0.866i)9-s + (−2.23 + 1.29i)11-s + (−3.34 − 1.33i)13-s + (−0.866 + 0.5i)15-s + (2.33 − 4.04i)17-s + (−4.86 − 2.80i)19-s − 3.81i·21-s + (−0.762 − 1.32i)23-s − 25-s − 0.999·27-s + (−2.52 − 4.37i)29-s + 10.3i·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + 0.447i·5-s + (−1.24 − 0.720i)7-s + (−0.166 + 0.288i)9-s + (−0.674 + 0.389i)11-s + (−0.928 − 0.371i)13-s + (−0.223 + 0.129i)15-s + (0.565 − 0.980i)17-s + (−1.11 − 0.643i)19-s − 0.832i·21-s + (−0.158 − 0.275i)23-s − 0.200·25-s − 0.192·27-s + (−0.469 − 0.812i)29-s + 1.86i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.793 + 0.608i$
Analytic conductor: \(6.22833\)
Root analytic conductor: \(2.49566\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{780} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 780,\ (\ :1/2),\ -0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0534508 - 0.157609i\)
\(L(\frac12)\) \(\approx\) \(0.0534508 - 0.157609i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.34 + 1.33i)T \)
good7 \( 1 + (3.30 + 1.90i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.23 - 1.29i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.33 + 4.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.86 + 2.80i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.762 + 1.32i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.52 + 4.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 + (-3.55 + 2.05i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.33 - 4.23i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.26 - 2.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.55iT - 47T^{2} \)
53 \( 1 - 5.16T + 53T^{2} \)
59 \( 1 + (0.501 + 0.289i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.97 + 6.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.235 - 0.136i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.5 + 6.66i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.9iT - 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 - 7.52iT - 83T^{2} \)
89 \( 1 + (4.37 - 2.52i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.76 - 3.32i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.12145423331581408029860751533, −9.375654073044230956722627464861, −8.248480066365316627528721499843, −7.22330239094698303659961606254, −6.70292472275678139024478846810, −5.39003267212072000260119239994, −4.42468475253459273806154638327, −3.27705861500012046885906057713, −2.52972543904079793458240696629, −0.07328923834183432787103100269, 1.96150976019529877994365317125, 3.01460620084082524290470821428, 4.14459547297200502787937165815, 5.61575474449066733745573079090, 6.12757046893215673859325378140, 7.23545446204621117776052069880, 8.151475565283864232854004519333, 8.889854325879300958215073735444, 9.729352910269050367268750631292, 10.42135562719684002449530562225

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