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Properties

Label 2-390-65.32-c1-0-8
Degree $2$
Conductor $390$
Sign $0.609 + 0.792i$
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 + (0.258 + 0.965i)3-s + (−0.499 − 0.866i)4-s + (−2.23 + 0.00342i)5-s + (0.965 + 0.258i)6-s + (3.25 − 1.88i)7-s − 0.999·8-s + (−0.866 + 0.499i)9-s + (−1.11 + 1.93i)10-s + (4.81 − 1.28i)11-s + (0.707 − 0.707i)12-s + (3.57 + 0.440i)13-s − 3.76i·14-s + (−0.582 − 2.15i)15-s + (−0.5 + 0.866i)16-s + (−5.59 − 1.49i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.149 + 0.557i)3-s + (−0.249 − 0.433i)4-s + (−0.999 + 0.00152i)5-s + (0.394 + 0.105i)6-s + (1.23 − 0.710i)7-s − 0.353·8-s + (−0.288 + 0.166i)9-s + (−0.352 + 0.612i)10-s + (1.45 − 0.388i)11-s + (0.204 − 0.204i)12-s + (0.992 + 0.122i)13-s − 1.00i·14-s + (−0.150 − 0.557i)15-s + (−0.125 + 0.216i)16-s + (−1.35 − 0.363i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.609 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.609 + 0.792i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.609 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49730 - 0.737096i\)
\(L(\frac12)\) \(\approx\) \(1.49730 - 0.737096i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (2.23 - 0.00342i)T \)
13 \( 1 + (-3.57 - 0.440i)T \)
good7 \( 1 + (-3.25 + 1.88i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.81 + 1.28i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (5.59 + 1.49i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.910 + 3.39i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-5.57 + 1.49i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.67 + 2.12i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.51 - 2.51i)T - 31iT^{2} \)
37 \( 1 + (-1.47 - 0.851i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.10 - 7.85i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.161 + 0.601i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 9.18iT - 47T^{2} \)
53 \( 1 + (6.03 - 6.03i)T - 53iT^{2} \)
59 \( 1 + (5.50 + 1.47i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.64 + 4.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.69 - 4.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.0 + 2.95i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 5.27T + 73T^{2} \)
79 \( 1 - 4.66iT - 79T^{2} \)
83 \( 1 + 0.863iT - 83T^{2} \)
89 \( 1 + (3.59 + 13.4i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-8.93 - 15.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.23683878820818652813766643872, −10.81829769783251361720778658634, −9.173717193059289037680758471404, −8.729001061148116038097410902450, −7.52026088586545265406418147614, −6.38493395854958916233875530411, −4.66153457799923736125804125689, −4.30761589724303808116218687066, −3.20018629245371824702192953144, −1.22317088276891034053016838835, 1.68469402690422282410284797657, 3.59390716174961738140163579202, 4.52087170371070583447558905875, 5.76459321895953347587573944335, 6.83972321637886006612734282617, 7.63790447562729668335958815834, 8.678421872068408984698193003753, 8.950282128197542703608690002342, 11.02827878654116191755755389872, 11.53968110100030161684231491685

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