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Properties

Label 2-390-65.49-c1-0-0
Degree $2$
Conductor $390$
Sign $-0.684 + 0.729i$
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  + (0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (−2.03 − 0.928i)5-s + (−0.866 − 0.499i)6-s + (−1.40 + 2.42i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−0.213 − 2.22i)10-s + (−0.515 + 0.297i)11-s − 0.999i·12-s + (1.10 − 3.43i)13-s − 2.80·14-s + (2.22 − 0.213i)15-s + (−0.5 − 0.866i)16-s + (−4.98 − 2.87i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.909 − 0.415i)5-s + (−0.353 − 0.204i)6-s + (−0.530 + 0.918i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.0673 − 0.703i)10-s + (−0.155 + 0.0896i)11-s − 0.288i·12-s + (0.307 − 0.951i)13-s − 0.749·14-s + (0.574 − 0.0550i)15-s + (−0.125 − 0.216i)16-s + (−1.20 − 0.697i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.684 + 0.729i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.684 + 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0507497 - 0.117259i\)
\(L(\frac12)\) \(\approx\) \(0.0507497 - 0.117259i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (2.03 + 0.928i)T \)
13 \( 1 + (-1.10 + 3.43i)T \)
good7 \( 1 + (1.40 - 2.42i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.515 - 0.297i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (4.98 + 2.87i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.59 + 3.80i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.02 - 2.32i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.26 + 2.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.59iT - 31T^{2} \)
37 \( 1 + (-5.18 - 8.98i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.98 - 2.87i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.67 - 2.12i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.89T + 47T^{2} \)
53 \( 1 + 13.8iT - 53T^{2} \)
59 \( 1 + (-8.40 - 4.85i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.41 - 5.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.93 - 6.80i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.11 + 0.642i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 + 1.83T + 79T^{2} \)
83 \( 1 + 4.19T + 83T^{2} \)
89 \( 1 + (5.24 - 3.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.45 + 14.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.86010629522467123858219433705, −11.25942998396260956926398066974, −10.03222554204323892740499311957, −8.845310178592011599826876629006, −8.311261521443972005131883228725, −7.01829501414055200826466005098, −6.14050850739098806887668460785, −5.08417758629269042737369186096, −4.23929261586401079590368925217, −2.88639650766730269469335268380, 0.07590948888695533562723495423, 2.13379120202962665539777427992, 3.98209037440555618101205201937, 4.20627864904202308432651844266, 6.09681658757003007213972122000, 6.74301601556649145400069927686, 7.85392228109841127281000322811, 8.955326725788848331113946965857, 10.34940713307947413694059748470, 10.80404473878573643224480883519

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