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Properties

Label 2-390-39.8-c1-0-8
Degree $2$
Conductor $390$
Sign $0.0100 + 0.999i$
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  + (0.707 − 0.707i)2-s + (−1.58 + 0.707i)3-s − 1.00i·4-s + (0.707 − 0.707i)5-s + (−0.618 + 1.61i)6-s + (−0.707 − 0.707i)8-s + (2.00 − 2.23i)9-s − 1.00i·10-s + (−1.41 − 1.41i)11-s + (0.707 + 1.58i)12-s + (0.418 − 3.58i)13-s + (−0.618 + 1.61i)15-s − 1.00·16-s + 7.30·17-s + (−0.166 − 2.99i)18-s + (−5.16 − 5.16i)19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.912 + 0.408i)3-s − 0.500i·4-s + (0.316 − 0.316i)5-s + (−0.252 + 0.660i)6-s + (−0.250 − 0.250i)8-s + (0.666 − 0.745i)9-s − 0.316i·10-s + (−0.426 − 0.426i)11-s + (0.204 + 0.456i)12-s + (0.116 − 0.993i)13-s + (−0.159 + 0.417i)15-s − 0.250·16-s + 1.77·17-s + (−0.0393 − 0.706i)18-s + (−1.18 − 1.18i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0100 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0100 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.0100 + 0.999i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.0100 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.922990 - 0.913715i\)
\(L(\frac12)\) \(\approx\) \(0.922990 - 0.913715i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (1.58 - 0.707i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (-0.418 + 3.58i)T \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 + (1.41 + 1.41i)T + 11iT^{2} \)
17 \( 1 - 7.30T + 17T^{2} \)
19 \( 1 + (5.16 + 5.16i)T + 19iT^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 + 4.47iT - 29T^{2} \)
31 \( 1 + (3 + 3i)T + 31iT^{2} \)
37 \( 1 + (4 - 4i)T - 37iT^{2} \)
41 \( 1 + (2.23 - 2.23i)T - 41iT^{2} \)
43 \( 1 + 7.16iT - 43T^{2} \)
47 \( 1 + (-7.30 - 7.30i)T + 47iT^{2} \)
53 \( 1 - 1.41iT - 53T^{2} \)
59 \( 1 + (-5.88 - 5.88i)T + 59iT^{2} \)
61 \( 1 - 10.6T + 61T^{2} \)
67 \( 1 + (-7.16 - 7.16i)T + 67iT^{2} \)
71 \( 1 + (3.87 - 3.87i)T - 71iT^{2} \)
73 \( 1 + (5.16 - 5.16i)T - 73iT^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + (5.65 - 5.65i)T - 83iT^{2} \)
89 \( 1 + (-0.592 - 0.592i)T + 89iT^{2} \)
97 \( 1 + (1.16 + 1.16i)T + 97iT^{2} \)
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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

−11.03914412592431235960138752826, −10.40411439058108747540856132978, −9.633224904450247911123497152313, −8.472360507043120608490265287414, −7.09053535459362278714767503668, −5.79411368697443863159892534351, −5.35226745065990645164123841948, −4.19205438058945417481379824350, −2.88958039341402657860979047727, −0.870711104604472515061738190478, 1.85444906351586787908921208794, 3.66669233351852035148126783434, 5.01402861586283134265230780542, 5.76186308852493955044577506115, 6.76623432235011500173781255142, 7.41828779932187412726318755162, 8.567732584471054116518430769618, 9.966773132790116787352393339158, 10.68421792836166594629874906010, 11.74219433624151396051328643669

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