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Properties

Label 2-390-195.77-c1-0-6
Degree $2$
Conductor $390$
Sign $-0.743 - 0.668i$
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  + (0.707 + 0.707i)2-s + (1.14 + 1.29i)3-s + 1.00i·4-s + (−2.23 + 0.00995i)5-s + (−0.106 + 1.72i)6-s + (−1.03 + 1.03i)7-s + (−0.707 + 0.707i)8-s + (−0.367 + 2.97i)9-s + (−1.58 − 1.57i)10-s − 3.70·11-s + (−1.29 + 1.14i)12-s + (1.20 + 3.39i)13-s − 1.45·14-s + (−2.57 − 2.89i)15-s − 1.00·16-s + (1.82 − 1.82i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.662 + 0.749i)3-s + 0.500i·4-s + (−0.999 + 0.00444i)5-s + (−0.0434 + 0.705i)6-s + (−0.389 + 0.389i)7-s + (−0.250 + 0.250i)8-s + (−0.122 + 0.992i)9-s + (−0.502 − 0.497i)10-s − 1.11·11-s + (−0.374 + 0.331i)12-s + (0.335 + 0.942i)13-s − 0.389·14-s + (−0.665 − 0.746i)15-s − 0.250·16-s + (0.442 − 0.442i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.743 - 0.668i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.743 - 0.668i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.574917 + 1.49973i\)
\(L(\frac12)\) \(\approx\) \(0.574917 + 1.49973i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-1.14 - 1.29i)T \)
5 \( 1 + (2.23 - 0.00995i)T \)
13 \( 1 + (-1.20 - 3.39i)T \)
good7 \( 1 + (1.03 - 1.03i)T - 7iT^{2} \)
11 \( 1 + 3.70T + 11T^{2} \)
17 \( 1 + (-1.82 + 1.82i)T - 17iT^{2} \)
19 \( 1 - 6.37T + 19T^{2} \)
23 \( 1 + (-3.59 - 3.59i)T + 23iT^{2} \)
29 \( 1 + 0.495T + 29T^{2} \)
31 \( 1 + 2.88iT - 31T^{2} \)
37 \( 1 + (-2.07 + 2.07i)T - 37iT^{2} \)
41 \( 1 + 1.65T + 41T^{2} \)
43 \( 1 + (-8.68 + 8.68i)T - 43iT^{2} \)
47 \( 1 + (-2.79 - 2.79i)T + 47iT^{2} \)
53 \( 1 + (5.90 + 5.90i)T + 53iT^{2} \)
59 \( 1 + 0.692iT - 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 + (6.79 - 6.79i)T - 67iT^{2} \)
71 \( 1 - 4.41T + 71T^{2} \)
73 \( 1 + (-8.02 - 8.02i)T + 73iT^{2} \)
79 \( 1 - 6.09iT - 79T^{2} \)
83 \( 1 + (-4.84 + 4.84i)T - 83iT^{2} \)
89 \( 1 + 2.66iT - 89T^{2} \)
97 \( 1 + (-6.98 + 6.98i)T - 97iT^{2} \)
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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

−11.64255832798017688673531346756, −10.88561165887501624440658399686, −9.596612953587768619346341683941, −8.893991881711124657688735187019, −7.78292036735970578336075472883, −7.28003368871719236605182749122, −5.64380741550640275328295570432, −4.75743341795182044329542177337, −3.64998318185175785811569932012, −2.79877061417315758218615988902, 0.880776059228128325575066724635, 2.89840420067719608014733106554, 3.46284565302149376259633178534, 4.90738570621967174285310441545, 6.19469948775061700447830417763, 7.46912244891215159556234396062, 7.924926638443764631089337850187, 9.080431360701664915031759917866, 10.26954964152604822344300559838, 11.03976952821142942666122520156

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