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Properties

Label 2-390-195.68-c1-0-17
Degree $2$
Conductor $390$
Sign $0.998 - 0.0471i$
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  + (−0.258 + 0.965i)2-s + (1.29 − 1.15i)3-s + (−0.866 − 0.499i)4-s + (1.49 − 1.66i)5-s + (0.776 + 1.54i)6-s + (1.04 + 3.88i)7-s + (0.707 − 0.707i)8-s + (0.350 − 2.97i)9-s + (1.22 + 1.87i)10-s + (2.62 − 1.51i)11-s + (−1.69 + 0.349i)12-s + (−3.59 + 0.269i)13-s − 4.02·14-s + (0.0160 − 3.87i)15-s + (0.500 + 0.866i)16-s + (1.60 − 0.429i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.747 − 0.664i)3-s + (−0.433 − 0.249i)4-s + (0.667 − 0.744i)5-s + (0.317 + 0.632i)6-s + (0.393 + 1.46i)7-s + (0.249 − 0.249i)8-s + (0.116 − 0.993i)9-s + (0.386 + 0.592i)10-s + (0.792 − 0.457i)11-s + (−0.489 + 0.100i)12-s + (−0.997 + 0.0747i)13-s − 1.07·14-s + (0.00414 − 0.999i)15-s + (0.125 + 0.216i)16-s + (0.389 − 0.104i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.998 - 0.0471i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.998 - 0.0471i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74790 + 0.0411943i\)
\(L(\frac12)\) \(\approx\) \(1.74790 + 0.0411943i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (-1.29 + 1.15i)T \)
5 \( 1 + (-1.49 + 1.66i)T \)
13 \( 1 + (3.59 - 0.269i)T \)
good7 \( 1 + (-1.04 - 3.88i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.62 + 1.51i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.60 + 0.429i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.154 - 0.0891i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.98 - 0.798i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.12 + 1.95i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.91T + 31T^{2} \)
37 \( 1 + (-3.53 - 0.946i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (6.86 - 3.96i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (10.6 - 2.85i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-6.89 - 6.89i)T + 47iT^{2} \)
53 \( 1 + (2.15 - 2.15i)T - 53iT^{2} \)
59 \( 1 + (-4.20 + 7.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.02 - 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.9 + 3.73i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (13.1 + 7.61i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.83 - 7.83i)T + 73iT^{2} \)
79 \( 1 - 6.38iT - 79T^{2} \)
83 \( 1 + (-3.82 + 3.82i)T - 83iT^{2} \)
89 \( 1 + (4.34 + 7.53i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.80 - 10.4i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.73010915653004304671281261440, −9.847442495327106748994875109820, −9.185207867486829149339268718131, −8.600676448392008978185417290402, −7.79066800749208298961986665964, −6.52847681294839187779780643844, −5.73113944583757297046650001562, −4.69063315141972863087462124135, −2.81158607955812232643893516014, −1.49023436347122006668153208858, 1.73798584214618288399711893420, 3.08018184506833659453021831190, 4.08437588330738054054529346572, 5.06905024255032935958256467123, 6.91640631342082081558431176395, 7.58795401150338198323358498897, 8.828200655201270746587390453868, 9.856178720763738547329890047639, 10.22380667012855852972707836210, 10.95191369038741135173672736720

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