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Properties

Label 2-390-195.77-c1-0-15
Degree $2$
Conductor $390$
Sign $0.992 + 0.126i$
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 + (−1.53 − 0.805i)3-s + 1.00i·4-s + (1.53 − 1.62i)5-s + (−0.514 − 1.65i)6-s + (−0.677 + 0.677i)7-s + (−0.707 + 0.707i)8-s + (1.70 + 2.47i)9-s + (2.23 − 0.0614i)10-s + 2.37·11-s + (0.805 − 1.53i)12-s + (0.799 − 3.51i)13-s − 0.957·14-s + (−3.66 + 1.25i)15-s − 1.00·16-s + (5.31 − 5.31i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.885 − 0.465i)3-s + 0.500i·4-s + (0.687 − 0.726i)5-s + (−0.209 − 0.675i)6-s + (−0.255 + 0.255i)7-s + (−0.250 + 0.250i)8-s + (0.567 + 0.823i)9-s + (0.706 − 0.0194i)10-s + 0.716·11-s + (0.232 − 0.442i)12-s + (0.221 − 0.975i)13-s − 0.255·14-s + (−0.946 + 0.323i)15-s − 0.250·16-s + (1.29 − 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.992 + 0.126i$
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: \((2,\ 390,\ (\ :1/2),\ 0.992 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54102 - 0.0975227i\)
\(L(\frac12)\) \(\approx\) \(1.54102 - 0.0975227i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.53 + 0.805i)T \)
5 \( 1 + (-1.53 + 1.62i)T \)
13 \( 1 + (-0.799 + 3.51i)T \)
good7 \( 1 + (0.677 - 0.677i)T - 7iT^{2} \)
11 \( 1 - 2.37T + 11T^{2} \)
17 \( 1 + (-5.31 + 5.31i)T - 17iT^{2} \)
19 \( 1 - 2.08T + 19T^{2} \)
23 \( 1 + (-1.90 - 1.90i)T + 23iT^{2} \)
29 \( 1 - 1.89T + 29T^{2} \)
31 \( 1 - 8.49iT - 31T^{2} \)
37 \( 1 + (2.64 - 2.64i)T - 37iT^{2} \)
41 \( 1 + 8.75T + 41T^{2} \)
43 \( 1 + (-0.132 + 0.132i)T - 43iT^{2} \)
47 \( 1 + (-2.08 - 2.08i)T + 47iT^{2} \)
53 \( 1 + (5.82 + 5.82i)T + 53iT^{2} \)
59 \( 1 + 11.2iT - 59T^{2} \)
61 \( 1 - 9.29T + 61T^{2} \)
67 \( 1 + (8.47 - 8.47i)T - 67iT^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + (3.69 + 3.69i)T + 73iT^{2} \)
79 \( 1 + 1.03iT - 79T^{2} \)
83 \( 1 + (0.475 - 0.475i)T - 83iT^{2} \)
89 \( 1 - 8.15iT - 89T^{2} \)
97 \( 1 + (-1.11 + 1.11i)T - 97iT^{2} \)
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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

−11.72764295329896885182296792816, −10.37157066174039131099134012724, −9.496078759387427979687797963692, −8.371324163973698046194838713563, −7.29722103517805323948819195832, −6.36949011925352923203190482583, −5.41425813998493431643568873685, −4.95349365375785204946118102829, −3.16298929631201187123691827257, −1.20614492226452982951392986495, 1.54904848447065725290225087046, 3.37042895769667088630981625004, 4.26570191990943832672697263426, 5.63539830382039676085284386855, 6.27662231822076129145393855320, 7.15782916286309961116359753220, 8.993236866997444690760380480516, 9.961594743077418201512994682375, 10.39339448049577576319348445899, 11.38933501426344505727338575770

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