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Properties

Label 2-390-13.10-c1-0-4
Degree $2$
Conductor $390$
Sign $-0.0575 + 0.998i$
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  + (−0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + i·5-s + (0.866 + 0.499i)6-s + (−2.01 − 1.16i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (4.62 − 2.67i)11-s − 0.999·12-s + (−3.60 − 0.161i)13-s + 2.32·14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (2 − 3.46i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + 0.447i·5-s + (0.353 + 0.204i)6-s + (−0.760 − 0.438i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.158 − 0.273i)10-s + (1.39 − 0.805i)11-s − 0.288·12-s + (−0.998 − 0.0447i)13-s + 0.620·14-s + (0.223 − 0.129i)15-s + (−0.125 − 0.216i)16-s + (0.485 − 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0575 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0575 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.0575 + 0.998i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.0575 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.438204 - 0.464187i\)
\(L(\frac12)\) \(\approx\) \(0.438204 - 0.464187i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 - iT \)
13 \( 1 + (3.60 + 0.161i)T \)
good7 \( 1 + (2.01 + 1.16i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.62 + 2.67i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.48 + 2.01i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.46 + 4.27i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.14 + 3.71i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.47iT - 31T^{2} \)
37 \( 1 + (2.72 - 1.57i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.29 + 1.32i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.12 + 10.6i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.81iT - 47T^{2} \)
53 \( 1 + 5.48T + 53T^{2} \)
59 \( 1 + (-5.87 - 3.39i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.267 - 0.464i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.55 + 2.05i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-13.7 - 7.93i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.5iT - 73T^{2} \)
79 \( 1 + 7.96T + 79T^{2} \)
83 \( 1 - 11.3iT - 83T^{2} \)
89 \( 1 + (1.50 - 0.869i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.9 + 8.05i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.05404530675080123861423755382, −10.03992066795710519503849349116, −9.310897121253665983972708778438, −8.233051329666976392613871718672, −7.07648625953978163540009577535, −6.64159951016448915596238939331, −5.64281106750578345018035309544, −4.00644010054907671790170415197, −2.47768687239125723375955357895, −0.53098147218661685358093019673, 1.74699202038698265764186127350, 3.45196510630318440799376819684, 4.49396203915352315725835463499, 5.88160268944065922153705434790, 6.83626395936685011702448432987, 8.033978352324885879293626570115, 9.258974106129526412127824891365, 9.539087061612929942551204450903, 10.44985645085181747432889634508, 11.56382937464807639015651639470

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