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Properties

Label 2-390-65.4-c1-0-2
Degree $2$
Conductor $390$
Sign $0.907 - 0.420i$
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 + (−2.10 + 0.767i)5-s + (0.866 − 0.499i)6-s + (−0.823 − 1.42i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (0.385 − 2.20i)10-s + (2.08 + 1.20i)11-s + 0.999i·12-s + (3.59 − 0.256i)13-s + 1.64·14-s + (2.20 + 0.385i)15-s + (−0.5 + 0.866i)16-s + (−0.210 + 0.121i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.939 + 0.343i)5-s + (0.353 − 0.204i)6-s + (−0.311 − 0.538i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (0.121 − 0.696i)10-s + (0.628 + 0.362i)11-s + 0.288i·12-s + (0.997 − 0.0710i)13-s + 0.439·14-s + (0.568 + 0.0994i)15-s + (−0.125 + 0.216i)16-s + (−0.0511 + 0.0295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.907 - 0.420i$
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: \((2,\ 390,\ (\ :1/2),\ 0.907 - 0.420i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.805920 + 0.177872i\)
\(L(\frac12)\) \(\approx\) \(0.805920 + 0.177872i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.10 - 0.767i)T \)
13 \( 1 + (-3.59 + 0.256i)T \)
good7 \( 1 + (0.823 + 1.42i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.08 - 1.20i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.210 - 0.121i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.82 + 2.20i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.46 - 4.31i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0221 + 0.0383i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.24iT - 31T^{2} \)
37 \( 1 + (4.47 - 7.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.210 - 0.121i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.82 + 3.36i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.29T + 47T^{2} \)
53 \( 1 + 2.44iT - 53T^{2} \)
59 \( 1 + (8.35 - 4.82i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.31 - 2.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.937 - 1.62i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.53 - 3.77i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.70T + 73T^{2} \)
79 \( 1 - 6.79T + 79T^{2} \)
83 \( 1 - 17.4T + 83T^{2} \)
89 \( 1 + (8.69 + 5.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.25 + 14.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.28824332726240865663117660409, −10.59664884144149571357234689676, −9.447360029860630098130039164795, −8.490329901418009049361537956319, −7.28938405407200258963213930554, −6.99371643842838496917392072436, −5.82512273122452454768326546142, −4.54397712600378348330910300687, −3.39480600873837809870063669586, −0.990231505264190338223272438423, 1.00743015795294230098171818119, 3.15054218299451989751892011311, 4.07355460359940142488101625961, 5.28011909592535702709929584008, 6.50800929664163162552653513331, 7.67909458471506783173622478470, 8.898961302677228720041378468529, 9.171225631265790259442830014225, 10.71510202944490492386089461507, 11.09513615190708290213017734161

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