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Properties

Label 2-390-65.33-c1-0-2
Degree $2$
Conductor $390$
Sign $-0.377 - 0.926i$
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  + (−0.5 + 0.866i)2-s + (−0.965 + 0.258i)3-s + (−0.499 − 0.866i)4-s + (−2.18 − 0.473i)5-s + (0.258 − 0.965i)6-s + (2.35 − 1.36i)7-s + 0.999·8-s + (0.866 − 0.499i)9-s + (1.50 − 1.65i)10-s + (0.843 + 3.14i)11-s + (0.707 + 0.707i)12-s + (1.21 + 3.39i)13-s + 2.72i·14-s + (2.23 − 0.108i)15-s + (−0.5 + 0.866i)16-s + (−1.25 + 4.70i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.557 + 0.149i)3-s + (−0.249 − 0.433i)4-s + (−0.977 − 0.211i)5-s + (0.105 − 0.394i)6-s + (0.890 − 0.514i)7-s + 0.353·8-s + (0.288 − 0.166i)9-s + (0.475 − 0.523i)10-s + (0.254 + 0.949i)11-s + (0.204 + 0.204i)12-s + (0.335 + 0.941i)13-s + 0.727i·14-s + (0.576 − 0.0278i)15-s + (−0.125 + 0.216i)16-s + (−0.305 + 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.377 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.377 - 0.926i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.377 - 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.394511 + 0.586553i\)
\(L(\frac12)\) \(\approx\) \(0.394511 + 0.586553i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (2.18 + 0.473i)T \)
13 \( 1 + (-1.21 - 3.39i)T \)
good7 \( 1 + (-2.35 + 1.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.843 - 3.14i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.25 - 4.70i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (5.26 + 1.40i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.42 - 5.30i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (6.99 + 4.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.70 - 3.70i)T + 31iT^{2} \)
37 \( 1 + (-7.40 - 4.27i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.05 + 1.89i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.885 - 0.237i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 8.15iT - 47T^{2} \)
53 \( 1 + (-3.97 - 3.97i)T + 53iT^{2} \)
59 \( 1 + (0.540 - 2.01i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.12 + 8.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.11 - 7.12i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.35 - 8.80i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 3.61T + 73T^{2} \)
79 \( 1 - 4.87iT - 79T^{2} \)
83 \( 1 + 13.6iT - 83T^{2} \)
89 \( 1 + (-2.29 + 0.616i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (5.57 + 9.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.26877435936314929926714492897, −10.95336625076088248126257577155, −9.681647287288733344082635347200, −8.703611254855631800014656134485, −7.77919248888544492586509331112, −7.04322674536902763021694644431, −5.99365287774734365052984297512, −4.44773054700560973453230579157, −4.25632024734272459348486957977, −1.51341812260914590667653321123, 0.61621220000099241997367600218, 2.53878029275212553156724779858, 3.91651881548929804290134878269, 5.02141296359836174417840382589, 6.24353907159674163740776146551, 7.55075579382567668211328467897, 8.298586768581226378665594748114, 9.065945780555521717622804785386, 10.59377639958245123978819010120, 11.08143255005985077671304659855

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