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Properties

Label 2-390-39.5-c1-0-7
Degree $2$
Conductor $390$
Sign $-0.315 - 0.948i$
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 + 1.41i)3-s + 1.00i·4-s + (0.707 + 0.707i)5-s + (−0.292 + 1.70i)6-s + (−0.707 + 0.707i)8-s + (−1.00 + 2.82i)9-s + 1.00i·10-s + (−1.41 + 1.41i)11-s + (−1.41 + 1.00i)12-s + (−3 − 2i)13-s + (−0.292 + 1.70i)15-s − 1.00·16-s + 4.24·17-s + (−2.70 + 1.29i)18-s + (6 − 6i)19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.577 + 0.816i)3-s + 0.500i·4-s + (0.316 + 0.316i)5-s + (−0.119 + 0.696i)6-s + (−0.250 + 0.250i)8-s + (−0.333 + 0.942i)9-s + 0.316i·10-s + (−0.426 + 0.426i)11-s + (−0.408 + 0.288i)12-s + (−0.832 − 0.554i)13-s + (−0.0756 + 0.440i)15-s − 0.250·16-s + 1.02·17-s + (−0.638 + 0.304i)18-s + (1.37 − 1.37i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.315 - 0.948i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.315 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22718 + 1.70213i\)
\(L(\frac12)\) \(\approx\) \(1.22718 + 1.70213i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.41i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (3 + 2i)T \)
good7 \( 1 + 7iT^{2} \)
11 \( 1 + (1.41 - 1.41i)T - 11iT^{2} \)
17 \( 1 - 4.24T + 17T^{2} \)
19 \( 1 + (-6 + 6i)T - 19iT^{2} \)
23 \( 1 - 1.41T + 23T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 + (3 - 3i)T - 31iT^{2} \)
37 \( 1 + (-5 - 5i)T + 37iT^{2} \)
41 \( 1 + (7.07 + 7.07i)T + 41iT^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + (-4.24 + 4.24i)T - 47iT^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 + (-2.82 + 2.82i)T - 59iT^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + (-5 + 5i)T - 67iT^{2} \)
71 \( 1 + (5.65 + 5.65i)T + 71iT^{2} \)
73 \( 1 + (-6 - 6i)T + 73iT^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (5.65 + 5.65i)T + 83iT^{2} \)
89 \( 1 + (4.24 - 4.24i)T - 89iT^{2} \)
97 \( 1 + (-10 + 10i)T - 97iT^{2} \)
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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

−11.60157003759171037686269303124, −10.47166448253875639492295470800, −9.780081550135617731072317801203, −8.875247154207263964967485541464, −7.70516523525033162017941841644, −7.06258620780537083217495270098, −5.40099739122978266624222503302, −4.97863658782715270985198834109, −3.47676681822620453935412557895, −2.59503202606996395796708432105, 1.28938405490221187358980306374, 2.61819772181064849607171230985, 3.71625485650797373136142951594, 5.24985340103808300097915671590, 6.09605847366919804460148865623, 7.39520917610058093275361626968, 8.139806958442382058356299822849, 9.443284150768076408336505446444, 9.929238864907929281287238367994, 11.36629238556856990658273666865

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