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Properties

Label 2-390-39.20-c1-0-13
Degree $2$
Conductor $390$
Sign $0.290 + 0.956i$
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  + (−0.258 − 0.965i)2-s + (1.46 − 0.926i)3-s + (−0.866 + 0.499i)4-s + (−0.707 + 0.707i)5-s + (−1.27 − 1.17i)6-s + (2.90 + 0.778i)7-s + (0.707 + 0.707i)8-s + (1.28 − 2.71i)9-s + (0.866 + 0.500i)10-s + (−0.227 + 0.0609i)11-s + (−0.803 + 1.53i)12-s + (2.63 − 2.46i)13-s − 3.00i·14-s + (−0.379 + 1.68i)15-s + (0.500 − 0.866i)16-s + (0.157 + 0.273i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.844 − 0.535i)3-s + (−0.433 + 0.249i)4-s + (−0.316 + 0.316i)5-s + (−0.520 − 0.479i)6-s + (1.09 + 0.294i)7-s + (0.249 + 0.249i)8-s + (0.427 − 0.904i)9-s + (0.273 + 0.158i)10-s + (−0.0686 + 0.0183i)11-s + (−0.232 + 0.442i)12-s + (0.730 − 0.683i)13-s − 0.803i·14-s + (−0.0979 + 0.436i)15-s + (0.125 − 0.216i)16-s + (0.0383 + 0.0663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.290 + 0.956i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.290 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34399 - 0.996126i\)
\(L(\frac12)\) \(\approx\) \(1.34399 - 0.996126i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (-1.46 + 0.926i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-2.63 + 2.46i)T \)
good7 \( 1 + (-2.90 - 0.778i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.227 - 0.0609i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.157 - 0.273i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.974 + 3.63i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.42 - 2.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.09 + 2.36i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.01 - 2.01i)T + 31iT^{2} \)
37 \( 1 + (-0.951 - 3.55i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.31 - 8.63i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (10.5 - 6.08i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.913 + 0.913i)T + 47iT^{2} \)
53 \( 1 + 8.08iT - 53T^{2} \)
59 \( 1 + (-1.51 + 5.65i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.17 - 8.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.5 - 2.82i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-12.3 - 3.29i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.383 + 0.383i)T - 73iT^{2} \)
79 \( 1 - 0.0348T + 79T^{2} \)
83 \( 1 + (-6.34 + 6.34i)T - 83iT^{2} \)
89 \( 1 + (8.55 - 2.29i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.65 - 13.6i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.38853010861005430781140457639, −10.22737785493314997830857988022, −9.238290271321583299017005415212, −8.193768249925446165782460649173, −7.890430931579205943343382739031, −6.56502158371091503248064202800, −5.05910070971787456260694422935, −3.74098593380577185372854307029, −2.68188358897036795008361356184, −1.37719861861223517640685160239, 1.74841732612292731270301275123, 3.73541937770568576230139960859, 4.53599890791637617250297741902, 5.59272985246476709976918860763, 7.10060352613208552138602888928, 7.994642372960925180058677452129, 8.554830422122916233427983012991, 9.409711648035617553889642663910, 10.47730098937642819722557333381, 11.27367673945420432993519878427

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