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Properties

Label 2-390-195.23-c1-0-7
Degree $2$
Conductor $390$
Sign $0.831 - 0.555i$
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 + (−0.793 + 1.53i)3-s + (0.866 + 0.499i)4-s + (−2.16 − 0.566i)5-s + (1.16 − 1.28i)6-s + (4.16 − 1.11i)7-s + (−0.707 − 0.707i)8-s + (−1.74 − 2.44i)9-s + (1.94 + 1.10i)10-s + (1.69 + 2.92i)11-s + (−1.45 + 0.936i)12-s + (−0.196 − 3.60i)13-s − 4.30·14-s + (2.58 − 2.88i)15-s + (0.500 + 0.866i)16-s + (2.12 − 0.569i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.458 + 0.888i)3-s + (0.433 + 0.249i)4-s + (−0.967 − 0.253i)5-s + (0.475 − 0.523i)6-s + (1.57 − 0.421i)7-s + (−0.249 − 0.249i)8-s + (−0.580 − 0.814i)9-s + (0.614 + 0.350i)10-s + (0.509 + 0.882i)11-s + (−0.420 + 0.270i)12-s + (−0.0543 − 0.998i)13-s − 1.15·14-s + (0.668 − 0.743i)15-s + (0.125 + 0.216i)16-s + (0.515 − 0.138i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.831 - 0.555i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.831 - 0.555i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.824266 + 0.249804i\)
\(L(\frac12)\) \(\approx\) \(0.824266 + 0.249804i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.793 - 1.53i)T \)
5 \( 1 + (2.16 + 0.566i)T \)
13 \( 1 + (0.196 + 3.60i)T \)
good7 \( 1 + (-4.16 + 1.11i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.69 - 2.92i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.12 + 0.569i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.88 - 5.00i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-8.91 - 2.38i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.535 - 0.927i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.90iT - 31T^{2} \)
37 \( 1 + (1.70 - 6.36i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-3.09 - 5.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.32 - 0.355i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.37 + 4.37i)T - 47iT^{2} \)
53 \( 1 + (-1.30 + 1.30i)T - 53iT^{2} \)
59 \( 1 + (1.13 + 0.654i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.42 + 9.39i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.54 + 5.78i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.45 + 5.98i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.13 + 2.13i)T - 73iT^{2} \)
79 \( 1 - 9.17iT - 79T^{2} \)
83 \( 1 + (10.5 + 10.5i)T + 83iT^{2} \)
89 \( 1 + (15.9 - 9.22i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.35 - 0.630i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.18981562825148949644313478740, −10.63296823172666672774066156960, −9.706014491775356371434722359392, −8.559733032356600099633348425583, −7.940534892462718477397950959115, −6.96082287814528116167108710938, −5.27777528718758242404564507490, −4.51233139821777256026269739573, −3.40103417897995674056788487648, −1.19699319136470025199790327536, 1.00145710477978904770983816991, 2.49589820973484247502004972210, 4.44731566195329717675422015324, 5.59817263194082048117134222260, 6.85759181747915496936392214567, 7.42981386707901962782994804001, 8.561727688503838460267415712043, 8.809689256882801656092806594852, 10.86550576329943493847803055462, 11.20285159320575053173491736573

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