↑

Properties

Label 2-780-12.11-c1-0-41
Degree $2$
Conductor $780$
Sign $0.989 + 0.144i$
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  + (−0.350 + 1.36i)2-s + (−0.604 − 1.62i)3-s + (−1.75 − 0.961i)4-s + i·5-s + (2.43 − 0.258i)6-s − 0.0304i·7-s + (1.93 − 2.06i)8-s + (−2.26 + 1.96i)9-s + (−1.36 − 0.350i)10-s + 1.47·11-s + (−0.500 + 3.42i)12-s − 13-s + (0.0417 + 0.0106i)14-s + (1.62 − 0.604i)15-s + (2.15 + 3.37i)16-s + 1.45i·17-s + ⋯
L(s)  = 1  + (−0.248 + 0.968i)2-s + (−0.348 − 0.937i)3-s + (−0.876 − 0.480i)4-s + 0.447i·5-s + (0.994 − 0.105i)6-s − 0.0115i·7-s + (0.683 − 0.730i)8-s + (−0.756 + 0.654i)9-s + (−0.433 − 0.110i)10-s + 0.443·11-s + (−0.144 + 0.989i)12-s − 0.277·13-s + (0.0111 + 0.00285i)14-s + (0.419 − 0.156i)15-s + (0.537 + 0.843i)16-s + 0.352i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.989 + 0.144i$
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: \((2,\ 780,\ (\ :1/2),\ 0.989 + 0.144i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00681 - 0.0731509i\)
\(L(\frac12)\) \(\approx\) \(1.00681 - 0.0731509i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.350 - 1.36i)T \)
3 \( 1 + (0.604 + 1.62i)T \)
5 \( 1 - iT \)
13 \( 1 + T \)
good7 \( 1 + 0.0304iT - 7T^{2} \)
11 \( 1 - 1.47T + 11T^{2} \)
17 \( 1 - 1.45iT - 17T^{2} \)
19 \( 1 + 3.38iT - 19T^{2} \)
23 \( 1 - 6.40T + 23T^{2} \)
29 \( 1 + 6.25iT - 29T^{2} \)
31 \( 1 + 5.55iT - 31T^{2} \)
37 \( 1 - 6.56T + 37T^{2} \)
41 \( 1 + 4.02iT - 41T^{2} \)
43 \( 1 + 10.7iT - 43T^{2} \)
47 \( 1 + 2.53T + 47T^{2} \)
53 \( 1 - 7.65iT - 53T^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 - 13.6T + 61T^{2} \)
67 \( 1 - 1.73iT - 67T^{2} \)
71 \( 1 - 3.60T + 71T^{2} \)
73 \( 1 + 8.38T + 73T^{2} \)
79 \( 1 + 7.35iT - 79T^{2} \)
83 \( 1 + 7.24T + 83T^{2} \)
89 \( 1 - 7.13iT - 89T^{2} \)
97 \( 1 - 9.19T + 97T^{2} \)
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

−10.20395954369366910788772396719, −9.184970212657618225996635144283, −8.400796923545329886223012646389, −7.41692111813850708266602127938, −6.94046686143995405905705206552, −6.08265237209154691431308694816, −5.28778519335622554686059647311, −4.06787247046715448532472750898, −2.40386285219779310805848132851, −0.74024920899607882209296479166, 1.11305110924134278968759300960, 2.84791657390984625504008651681, 3.81869129391680494831208084404, 4.77244269850715725912065107462, 5.44466527734464154400167195832, 6.84789823962760394199364382145, 8.193011911166513352435968611984, 8.931119024856113027372767623702, 9.613872138637162840597750548702, 10.26640191015710266715126407649

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