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Properties

Label 2-390-65.9-c1-0-13
Degree $2$
Conductor $390$
Sign $0.923 + 0.382i$
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 + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (1.23 − 1.86i)5-s + (−0.499 − 0.866i)6-s + (0.633 − 0.366i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (2 − i)10-s + (1.36 − 2.36i)11-s − 0.999i·12-s + (−1.59 − 3.23i)13-s + 0.732·14-s + (−2 + i)15-s + (−0.5 + 0.866i)16-s + (2.13 − 1.23i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.550 − 0.834i)5-s + (−0.204 − 0.353i)6-s + (0.239 − 0.138i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.632 − 0.316i)10-s + (0.411 − 0.713i)11-s − 0.288i·12-s + (−0.443 − 0.896i)13-s + 0.195·14-s + (−0.516 + 0.258i)15-s + (−0.125 + 0.216i)16-s + (0.517 − 0.298i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80228 - 0.358373i\)
\(L(\frac12)\) \(\approx\) \(1.80228 - 0.358373i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-1.23 + 1.86i)T \)
13 \( 1 + (1.59 + 3.23i)T \)
good7 \( 1 + (-0.633 + 0.366i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.36 + 2.36i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.13 + 1.23i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.36 - 4.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.63 - 2.09i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.232 + 0.401i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (5.13 + 2.96i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.598 - 1.03i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.83 - 3.36i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.66iT - 47T^{2} \)
53 \( 1 - 4.26iT - 53T^{2} \)
59 \( 1 + (-4.19 - 7.26i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.06 + 12.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.36 + 4.83i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.36 + 4.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.6iT - 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 - 8.73iT - 83T^{2} \)
89 \( 1 + (-4.46 + 7.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.66 + 5i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.52859133944278523699223606931, −10.43249093534051774237827475184, −9.423994266660018990443265485268, −8.262733023524602418998763385469, −7.48172243868375387871380818157, −6.17433218108565313884331912840, −5.50040577500015953509313098496, −4.65192971937851753970201915713, −3.14543059982715082473191192345, −1.26775948270686756842543698650, 1.85688672543914951850933127166, 3.20279883338899669632961126412, 4.54732101826440316193153756058, 5.39932122297646622323811700157, 6.62908784082645664500683189537, 7.12715649430882503629909929381, 8.907507789620671258497767469692, 9.909102532323296095177306870730, 10.46030274678265477586886335895, 11.59916252052081082997202787138

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