↑

Properties

Label 2-390-13.10-c1-0-5
Degree $2$
Conductor $390$
Sign $0.967 + 0.252i$
Analytic cond. $3.11416$
Root an. cond. $1.76469$
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  + (0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s − i·5-s + (0.866 + 0.499i)6-s + (2.59 + 1.5i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.232 − 0.133i)11-s + 0.999·12-s + (0.866 − 3.5i)13-s + 3·14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−2 + 3.46i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s − 0.447i·5-s + (0.353 + 0.204i)6-s + (0.981 + 0.566i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.158 − 0.273i)10-s + (0.0699 − 0.0403i)11-s + 0.288·12-s + (0.240 − 0.970i)13-s + 0.801·14-s + (0.223 − 0.129i)15-s + (−0.125 − 0.216i)16-s + (−0.485 + 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.967 + 0.252i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.967 + 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22529 - 0.285711i\)
\(L(\frac12)\) \(\approx\) \(2.22529 - 0.285711i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + iT \)
13 \( 1 + (-0.866 + 3.5i)T \)
good7 \( 1 + (-2.59 - 1.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.232 + 0.133i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.96 - 2.86i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.73 + 3i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.732 + 1.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.92iT - 31T^{2} \)
37 \( 1 + (5.13 - 2.96i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.46 - 2i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3 - 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.46iT - 47T^{2} \)
53 \( 1 + 0.267T + 53T^{2} \)
59 \( 1 + (9.92 + 5.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.267 + 0.464i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.26 + 0.732i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.1 + 6.46i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 - 3.07T + 79T^{2} \)
83 \( 1 + 9.46iT - 83T^{2} \)
89 \( 1 + (12.2 - 7.06i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.26 - 4.19i)T + (48.5 + 84.0i)T^{2} \)
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

−11.34813773277616399662528045856, −10.48392115903524503805349868572, −9.577735818600155130128289092603, −8.453804232535941062009265018857, −7.84676536867782507686165979142, −6.10766380145351258923259747093, −5.25569621345798412674428023677, −4.36347422254600825092272138870, −3.15495164670683074847833400087, −1.70166073836535242231125634679, 1.76340583897817762083979213066, 3.24575906275120943923344625320, 4.46392414259883872720695165795, 5.50322319980367032209801563753, 7.00738304865843536075069543736, 7.18238653535402705825599364545, 8.392223695706725375480649769779, 9.353354235762821197063387731416, 10.72297390958472045962545013325, 11.56008294835111656771740733631

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