Properties

Label 2-390-195.59-c1-0-22
Degree $2$
Conductor $390$
Sign $0.926 + 0.377i$
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 + (1.04 + 1.38i)3-s + (0.866 − 0.499i)4-s + (−0.189 − 2.22i)5-s + (1.36 + 1.06i)6-s + (0.885 − 3.30i)7-s + (0.707 − 0.707i)8-s + (−0.829 + 2.88i)9-s + (−0.759 − 2.10i)10-s + (0.0966 − 0.0259i)11-s + (1.59 + 0.677i)12-s + (2.78 − 2.29i)13-s − 3.42i·14-s + (2.88 − 2.58i)15-s + (0.500 − 0.866i)16-s + (−3.13 + 1.81i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.601 + 0.798i)3-s + (0.433 − 0.249i)4-s + (−0.0847 − 0.996i)5-s + (0.557 + 0.435i)6-s + (0.334 − 1.24i)7-s + (0.249 − 0.249i)8-s + (−0.276 + 0.961i)9-s + (−0.240 − 0.665i)10-s + (0.0291 − 0.00781i)11-s + (0.460 + 0.195i)12-s + (0.771 − 0.636i)13-s − 0.914i·14-s + (0.745 − 0.667i)15-s + (0.125 − 0.216i)16-s + (−0.760 + 0.439i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.926 + 0.377i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.926 + 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37218 - 0.464559i\)
\(L(\frac12)\) \(\approx\) \(2.37218 - 0.464559i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.04 - 1.38i)T \)
5 \( 1 + (0.189 + 2.22i)T \)
13 \( 1 + (-2.78 + 2.29i)T \)
good7 \( 1 + (-0.885 + 3.30i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.0966 + 0.0259i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (3.13 - 1.81i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.12 - 7.92i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.13 - 0.652i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.17 - 1.25i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.96 - 1.96i)T + 31iT^{2} \)
37 \( 1 + (2.84 - 0.763i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.87 - 6.98i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.74 - 3.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.71 + 4.71i)T - 47iT^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + (-2.21 + 8.27i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.57 + 9.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.50 + 5.62i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.701 + 0.187i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (2.39 + 2.39i)T + 73iT^{2} \)
79 \( 1 + 4.53T + 79T^{2} \)
83 \( 1 + (-12.5 - 12.5i)T + 83iT^{2} \)
89 \( 1 + (-6.88 + 1.84i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (5.08 + 1.36i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.01720964798769501607542949253, −10.52288342893578729571894462635, −9.574124756658023118050313142493, −8.349223846133184974967777870498, −7.85092865982955484665845456224, −6.25058718501434154928810029806, −5.03218015988457327860276603783, −4.20166857343942890036403950501, −3.48564845907043020527508609015, −1.56132803699559940698467350902, 2.21770076986453296793058773779, 2.88866418017845456979528067958, 4.34535749857252859621041762078, 5.85457445824431834010170606880, 6.64376030651387964453897848431, 7.36079437295842903033015006728, 8.627235742765001243302616534425, 9.153562674240133084011990373826, 10.86239717395185656645630050451, 11.55399678319517338999020686679

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