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Properties

Label 2-390-39.32-c1-0-5
Degree $2$
Conductor $390$
Sign $0.961 - 0.276i$
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 + (−1.39 + 1.02i)3-s + (0.866 − 0.499i)4-s + (−0.707 − 0.707i)5-s + (1.08 − 1.35i)6-s + (−0.213 + 0.796i)7-s + (−0.707 + 0.707i)8-s + (0.890 − 2.86i)9-s + (0.866 + 0.500i)10-s + (−0.981 − 3.66i)11-s + (−0.694 + 1.58i)12-s + (3.42 + 1.12i)13-s − 0.825i·14-s + (1.71 + 0.260i)15-s + (0.500 − 0.866i)16-s + (1.49 + 2.59i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.805 + 0.592i)3-s + (0.433 − 0.249i)4-s + (−0.316 − 0.316i)5-s + (0.441 − 0.552i)6-s + (−0.0807 + 0.301i)7-s + (−0.249 + 0.249i)8-s + (0.296 − 0.954i)9-s + (0.273 + 0.158i)10-s + (−0.295 − 1.10i)11-s + (−0.200 + 0.458i)12-s + (0.950 + 0.310i)13-s − 0.220i·14-s + (0.442 + 0.0671i)15-s + (0.125 − 0.216i)16-s + (0.363 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.961 - 0.276i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.961 - 0.276i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.718909 + 0.101325i\)
\(L(\frac12)\) \(\approx\) \(0.718909 + 0.101325i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.39 - 1.02i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-3.42 - 1.12i)T \)
good7 \( 1 + (0.213 - 0.796i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.981 + 3.66i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.49 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.17 - 0.314i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.76 + 3.05i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.48 - 3.74i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.00278 + 0.00278i)T - 31iT^{2} \)
37 \( 1 + (-10.9 + 2.94i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-4.61 + 1.23i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.40 + 1.96i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.01 - 7.01i)T - 47iT^{2} \)
53 \( 1 - 4.90iT - 53T^{2} \)
59 \( 1 + (-4.67 - 1.25i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (6.87 + 11.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.00 + 11.2i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.52 + 5.70i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (8.49 + 8.49i)T + 73iT^{2} \)
79 \( 1 - 5.90T + 79T^{2} \)
83 \( 1 + (5.36 + 5.36i)T + 83iT^{2} \)
89 \( 1 + (-1.85 - 6.93i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.65 + 2.05i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.99035206543592586551028009299, −10.70475010633217922042286265814, −9.426306770017688000233613984846, −8.738099678109660549897086962059, −7.81702782815679956949300922932, −6.35911558067616941598657932524, −5.82951473612353673105321419960, −4.54838064280233612736379773327, −3.22894635033445820818994389107, −0.928416017989031761318082907216, 1.04389544196956505640530811493, 2.68381287352581343110513226936, 4.34601817139093429071976866065, 5.68648444150821648698602570184, 6.79863225743901372875807871948, 7.48025321584049340385750107316, 8.292785421378184827616578627170, 9.734343484981923568727758202523, 10.36717464318171521623955833371, 11.36261598737597341994594216332

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