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Properties

Label 2-390-39.11-c1-0-6
Degree $2$
Conductor $390$
Sign $0.665 - 0.746i$
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 + (−1.05 − 1.37i)3-s + (0.866 + 0.499i)4-s + (−0.707 + 0.707i)5-s + (−0.667 − 1.59i)6-s + (1.27 + 4.74i)7-s + (0.707 + 0.707i)8-s + (−0.760 + 2.90i)9-s + (−0.866 + 0.500i)10-s + (0.174 − 0.649i)11-s + (−0.230 − 1.71i)12-s + (0.528 + 3.56i)13-s + 4.91i·14-s + (1.71 + 0.221i)15-s + (0.500 + 0.866i)16-s + (3.91 − 6.77i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.610 − 0.791i)3-s + (0.433 + 0.249i)4-s + (−0.316 + 0.316i)5-s + (−0.272 − 0.652i)6-s + (0.480 + 1.79i)7-s + (0.249 + 0.249i)8-s + (−0.253 + 0.967i)9-s + (−0.273 + 0.158i)10-s + (0.0524 − 0.195i)11-s + (−0.0666 − 0.495i)12-s + (0.146 + 0.989i)13-s + 1.31i·14-s + (0.443 + 0.0571i)15-s + (0.125 + 0.216i)16-s + (0.948 − 1.64i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.665 - 0.746i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.665 - 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48634 + 0.666482i\)
\(L(\frac12)\) \(\approx\) \(1.48634 + 0.666482i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.05 + 1.37i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-0.528 - 3.56i)T \)
good7 \( 1 + (-1.27 - 4.74i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.174 + 0.649i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-3.91 + 6.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.01 + 0.271i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.67 - 2.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.85 - 3.38i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.25 - 1.25i)T + 31iT^{2} \)
37 \( 1 + (-2.23 - 0.597i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (7.70 + 2.06i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.63 + 1.52i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.77 - 4.77i)T + 47iT^{2} \)
53 \( 1 + 13.3iT - 53T^{2} \)
59 \( 1 + (-0.859 + 0.230i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.16 + 5.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.47 + 5.49i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.56 - 13.3i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-6.70 + 6.70i)T - 73iT^{2} \)
79 \( 1 + 6.32T + 79T^{2} \)
83 \( 1 + (-3.44 + 3.44i)T - 83iT^{2} \)
89 \( 1 + (-2.97 + 11.1i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-10.5 + 2.83i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.69922675976960403085712095811, −11.16582996423125450896325954918, −9.506997078458857648216399907661, −8.473277615752320664541865723426, −7.45446684865607365222413001276, −6.61745976463064675807979307835, −5.51577354832608739710106293426, −5.02190697565845546715208678978, −3.13204551610985315313724879505, −1.93568550697422260082397924044, 1.03605190146335846958149703786, 3.56542975911442150098538703352, 4.12903229091127952366098128460, 5.12803781161468778161508924277, 6.14850306747177827072613703692, 7.38155937438283627835994075754, 8.268472337832233441592690164753, 9.870156628547318103087480106384, 10.50446998433516149653904493314, 11.02212383531061754430465086595

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