Properties

Label 2-390-13.4-c1-0-1
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: $\chi_{390} (121, \cdot )$
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

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

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