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Properties

Label 2-390-39.32-c1-0-4
Degree $2$
Conductor $390$
Sign $0.164 - 0.986i$
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  + (−0.965 + 0.258i)2-s + (0.888 + 1.48i)3-s + (0.866 − 0.499i)4-s + (−0.707 − 0.707i)5-s + (−1.24 − 1.20i)6-s + (0.431 − 1.60i)7-s + (−0.707 + 0.707i)8-s + (−1.42 + 2.64i)9-s + (0.866 + 0.500i)10-s + (0.985 + 3.67i)11-s + (1.51 + 0.843i)12-s + (2.83 + 2.22i)13-s + 1.66i·14-s + (0.422 − 1.67i)15-s + (0.500 − 0.866i)16-s + (0.660 + 1.14i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.513 + 0.858i)3-s + (0.433 − 0.249i)4-s + (−0.316 − 0.316i)5-s + (−0.507 − 0.492i)6-s + (0.162 − 0.608i)7-s + (−0.249 + 0.249i)8-s + (−0.473 + 0.880i)9-s + (0.273 + 0.158i)10-s + (0.297 + 1.10i)11-s + (0.436 + 0.243i)12-s + (0.786 + 0.617i)13-s + 0.445i·14-s + (0.109 − 0.433i)15-s + (0.125 − 0.216i)16-s + (0.160 + 0.277i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.164 - 0.986i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.164 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.867582 + 0.734637i\)
\(L(\frac12)\) \(\approx\) \(0.867582 + 0.734637i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.888 - 1.48i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-2.83 - 2.22i)T \)
good7 \( 1 + (-0.431 + 1.60i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.985 - 3.67i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.660 - 1.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.54 - 1.48i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.32 - 5.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.32 + 0.766i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.84 + 5.84i)T - 31iT^{2} \)
37 \( 1 + (9.72 - 2.60i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-4.33 + 1.16i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.30 - 3.63i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.18 + 9.18i)T - 47iT^{2} \)
53 \( 1 - 0.302iT - 53T^{2} \)
59 \( 1 + (3.71 + 0.995i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.23 - 3.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0586 + 0.218i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.95 + 14.7i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.682 + 0.682i)T + 73iT^{2} \)
79 \( 1 + 8.02T + 79T^{2} \)
83 \( 1 + (8.80 + 8.80i)T + 83iT^{2} \)
89 \( 1 + (-2.31 - 8.65i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.63 + 0.437i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.38833317028613521432887861909, −10.26497930634465989351694383670, −9.709075956588731835932027294693, −8.866675816571378945187403827857, −7.900299541828671475108904088263, −7.19416137001246906688014570104, −5.70932477115108403322433277285, −4.45709120121325605589126187826, −3.55209455736182209978357508374, −1.70057820974117838562865772421, 0.985781780172079773141440003328, 2.65661612949843574928786934401, 3.52301586551535938910392707156, 5.62547834866833916827862492978, 6.57278150353496017362681425881, 7.56400885679579369299650999427, 8.514677653005586851679799101877, 8.861871550893787501441183285990, 10.19791568634788407896700525652, 11.21749822611031462309282944128

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