Properties

Label 2-390-65.7-c1-0-0
Degree $2$
Conductor $390$
Sign $0.263 - 0.964i$
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 + (−0.258 − 0.965i)3-s + (0.499 + 0.866i)4-s + (−1.54 + 1.61i)5-s + (0.258 − 0.965i)6-s + (0.954 + 1.65i)7-s + 0.999i·8-s + (−0.866 + 0.499i)9-s + (−2.14 + 0.625i)10-s + (0.562 + 2.09i)11-s + (0.707 − 0.707i)12-s + (2.33 + 2.75i)13-s + 1.90i·14-s + (1.96 + 1.07i)15-s + (−0.5 + 0.866i)16-s + (−0.597 − 0.160i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.149 − 0.557i)3-s + (0.249 + 0.433i)4-s + (−0.691 + 0.722i)5-s + (0.105 − 0.394i)6-s + (0.360 + 0.625i)7-s + 0.353i·8-s + (−0.288 + 0.166i)9-s + (−0.678 + 0.197i)10-s + (0.169 + 0.633i)11-s + (0.204 − 0.204i)12-s + (0.646 + 0.762i)13-s + 0.510i·14-s + (0.506 + 0.277i)15-s + (−0.125 + 0.216i)16-s + (−0.144 − 0.0388i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.263 - 0.964i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.263 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30462 + 0.996301i\)
\(L(\frac12)\) \(\approx\) \(1.30462 + 0.996301i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (1.54 - 1.61i)T \)
13 \( 1 + (-2.33 - 2.75i)T \)
good7 \( 1 + (-0.954 - 1.65i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.562 - 2.09i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.597 + 0.160i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.46 - 0.927i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.653 - 0.175i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.93 + 1.69i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.691 - 0.691i)T + 31iT^{2} \)
37 \( 1 + (-1.42 + 2.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.83 + 2.09i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.901 - 3.36i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + (1.54 - 1.54i)T - 53iT^{2} \)
59 \( 1 + (-3.75 + 14.0i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.85 + 4.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.07 - 2.92i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.78 - 6.64i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 2.15iT - 73T^{2} \)
79 \( 1 + 5.11iT - 79T^{2} \)
83 \( 1 + 3.84T + 83T^{2} \)
89 \( 1 + (-0.804 + 0.215i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-15.8 + 9.16i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.57509979731527667939656278458, −11.09170719838314436133865603635, −9.629856751999293436273775640072, −8.417744162703998625480410990479, −7.57805630121181302133418787815, −6.76393776315258037454314419809, −5.89064386744791690782818654844, −4.63712317041162301474254717729, −3.46862361668157110424213854146, −2.07385929703355323217538245240, 0.999225838513310593171628839774, 3.23977752423170540651702040217, 4.10888949798702989035699935259, 5.04628447247929584882977555491, 5.99606628984974526688938850938, 7.45001271405898834862997580038, 8.391393504115563886854442066007, 9.382103814131145031663482359188, 10.51124390334064627206470911037, 11.22438044431778019762083290846

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