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Properties

Label 2-390-65.9-c1-0-15
Degree $2$
Conductor $390$
Sign $-0.974 + 0.222i$
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 + (−1.5 − 1.65i)5-s + (0.499 + 0.866i)6-s + (2.00 − 1.15i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (0.469 + 2.18i)10-s + (−0.158 + 0.274i)11-s − 0.999i·12-s + (0.866 − 3.5i)13-s − 2.31·14-s + (0.469 + 2.18i)15-s + (−0.5 + 0.866i)16-s + (−4.87 + 2.81i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.670 − 0.741i)5-s + (0.204 + 0.353i)6-s + (0.758 − 0.437i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.148 + 0.691i)10-s + (−0.0477 + 0.0826i)11-s − 0.288i·12-s + (0.240 − 0.970i)13-s − 0.619·14-s + (0.121 + 0.564i)15-s + (−0.125 + 0.216i)16-s + (−1.18 + 0.683i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.974 + 0.222i$
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: \((2,\ 390,\ (\ :1/2),\ -0.974 + 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0544730 - 0.483157i\)
\(L(\frac12)\) \(\approx\) \(0.0544730 - 0.483157i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 1.65i)T \)
13 \( 1 + (-0.866 + 3.5i)T \)
good7 \( 1 + (-2.00 + 1.15i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.158 - 0.274i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (4.87 - 2.81i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.15 + 2.00i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.20 + 4.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (4.87 + 2.81i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.65 + 6.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.73 - 2.15i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.31iT - 47T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 + (-4.31 - 7.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.65 + 6.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.2 - 6.47i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.841 + 1.45i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.31iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 1.68iT - 83T^{2} \)
89 \( 1 + (-7.63 + 13.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.19 + 3i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.94514912663466757603324107959, −10.23099219741886844844922078934, −8.840311322170869096429226079903, −8.170980184081625116335208806127, −7.43749997192508067568612777537, −6.18266634091115303705800004827, −4.84028236311073486897153651876, −3.90478576317493358920774035989, −1.95617626393175990760011811197, −0.40751719853749805095678692506, 2.03887873891635708420536567741, 3.85054685447367894905492840334, 5.00449281751034190794334067397, 6.23743364609573257068562946743, 7.03898247984701439894675590241, 8.055288983623615722458283023052, 8.913266703129015895062738421351, 9.955771288452721232148440370709, 10.93429885290118905907770061404, 11.50969193631650881922753137205

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