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Properties

Label 2-390-65.4-c1-0-6
Degree $2$
Conductor $390$
Sign $0.876 - 0.480i$
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 + (−0.230 − 2.22i)5-s + (−0.866 + 0.499i)6-s + (0.432 + 0.749i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (2.04 + 0.912i)10-s + (0.151 + 0.0874i)11-s − 0.999i·12-s + (1.35 + 3.34i)13-s − 0.865·14-s + (0.912 − 2.04i)15-s + (−0.5 + 0.866i)16-s + (7.08 − 4.08i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.103 − 0.994i)5-s + (−0.353 + 0.204i)6-s + (0.163 + 0.283i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (0.645 + 0.288i)10-s + (0.0456 + 0.0263i)11-s − 0.288i·12-s + (0.375 + 0.926i)13-s − 0.231·14-s + (0.235 − 0.527i)15-s + (−0.125 + 0.216i)16-s + (1.71 − 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.876 - 0.480i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.876 - 0.480i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34670 + 0.344776i\)
\(L(\frac12)\) \(\approx\) \(1.34670 + 0.344776i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.230 + 2.22i)T \)
13 \( 1 + (-1.35 - 3.34i)T \)
good7 \( 1 + (-0.432 - 0.749i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.151 - 0.0874i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-7.08 + 4.08i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.20 + 3.00i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.52 - 1.45i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.24 + 5.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.95iT - 31T^{2} \)
37 \( 1 + (0.879 - 1.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.08 + 4.08i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.94 - 4.58i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 - 2.48iT - 53T^{2} \)
59 \( 1 + (6.09 - 3.51i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.98 - 6.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.36 + 2.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-12.2 + 7.08i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 - 9.48T + 79T^{2} \)
83 \( 1 - 0.139T + 83T^{2} \)
89 \( 1 + (11.3 + 6.56i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.32 + 7.48i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.55104639175258602094611653026, −10.00721355458916193253340965143, −9.416920103408773198606021884345, −8.640201575566107001403152231311, −7.84842766838132594811917765035, −6.85340319574385666746698393450, −5.38249866225056436617686732349, −4.78316311468188923389726483347, −3.29790949815451012322332123848, −1.32726594854625284861379528582, 1.41903600236174698500313831235, 3.09196715857896594389506906372, 3.61222649285003970088733692278, 5.45048192293588107124121828585, 6.72828082226558426800795607855, 7.84353840453422090100801366456, 8.220753610828735787583180310809, 9.753058906806265222395738569318, 10.23760988476097393584608475391, 11.13118353556782443049588906516

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