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Properties

Label 2-390-195.17-c1-0-5
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: $\chi_{390} (17, \cdot )$
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

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

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