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Properties

Label 2-390-65.28-c1-0-10
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: $\chi_{390} (223, \cdot )$
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

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

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