Properties

Label 2-390-13.10-c1-0-7
Degree $2$
Conductor $390$
Sign $-0.790 + 0.612i$
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 + (−3.15 − 1.82i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−1.44 + 0.834i)11-s − 0.999·12-s + (−2.24 − 2.82i)13-s − 3.64·14-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 + (−1.19 − 0.688i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.158 − 0.273i)10-s + (−0.435 + 0.251i)11-s − 0.288·12-s + (−0.622 − 0.782i)13-s − 0.974·14-s + (−0.223 + 0.129i)15-s + (−0.125 − 0.216i)16-s + (0.485 − 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.790 + 0.612i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.790 + 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.417548 - 1.22129i\)
\(L(\frac12)\) \(\approx\) \(0.417548 - 1.22129i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.24 + 2.82i)T \)
good7 \( 1 + (3.15 + 1.82i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.44 - 0.834i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.46 - 3.15i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.622 - 1.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.02 + 8.69i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.21iT - 31T^{2} \)
37 \( 1 + (-8.54 + 4.93i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.04 + 4.64i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.78 - 6.55i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.82iT - 47T^{2} \)
53 \( 1 + 0.848T + 53T^{2} \)
59 \( 1 + (-5.29 - 3.05i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.73 - 6.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.7 + 7.37i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.04 + 1.75i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.2iT - 73T^{2} \)
79 \( 1 - 9.93T + 79T^{2} \)
83 \( 1 - 7.95iT - 83T^{2} \)
89 \( 1 + (-5.15 + 2.97i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.38 + 1.37i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.13171004964262379680274413650, −9.924891675465512960514202154474, −9.619080158313605159015289440456, −7.79915068734206422096268858493, −7.20736233142040021411858140727, −5.93981229811586573720103823493, −5.18510338374595966301721360070, −3.79991913214436395242998776301, −2.64847969836570994649287178487, −0.71739903756515032654615774137, 2.70292389482671878118778004262, 3.60490133179593382254443419732, 4.97478072078585111772673568201, 5.91202934099138669217756490412, 6.70035390906249320697684084824, 7.74637595746882709895524472369, 9.163549428521175155005288268792, 9.741849992200058705436132437549, 10.90188696963687819881093083474, 11.72001362589286004751118361891

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