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Properties

Label 2-390-13.4-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

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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: $\chi_{390} (121, \cdot )$
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 \( 1 + (-7.26 + 4.19i)T + (48.5 - 84.0i)T^{2} \)
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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

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

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