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Properties

Label 2-390-39.5-c1-0-9
Degree $2$
Conductor $390$
Sign $0.970 + 0.240i$
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 + (3.58 + 0.418i)13-s + (−0.618 − 1.61i)15-s − 1.00·16-s + 1.64·17-s + (0.166 − 2.99i)18-s + (1.16 − 1.16i)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 + (0.993 + 0.116i)13-s + (−0.159 − 0.417i)15-s − 0.250·16-s + 0.398·17-s + (0.0393 − 0.706i)18-s + (0.266 − 0.266i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.970 + 0.240i$
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: \((2,\ 390,\ (\ :1/2),\ 0.970 + 0.240i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45095 - 0.177139i\)
\(L(\frac12)\) \(\approx\) \(1.45095 - 0.177139i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.58 - 0.418i)T \)
good7 \( 1 + 7iT^{2} \)
11 \( 1 + (-1.41 + 1.41i)T - 11iT^{2} \)
17 \( 1 - 1.64T + 17T^{2} \)
19 \( 1 + (-1.16 + 1.16i)T - 19iT^{2} \)
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 \( 1 - 0.837iT - 43T^{2} \)
47 \( 1 + (-1.64 + 1.64i)T - 47iT^{2} \)
53 \( 1 - 1.41iT - 53T^{2} \)
59 \( 1 + (-3.05 + 3.05i)T - 59iT^{2} \)
61 \( 1 + 14.6T + 61T^{2} \)
67 \( 1 + (-0.837 + 0.837i)T - 67iT^{2} \)
71 \( 1 + (9.53 + 9.53i)T + 71iT^{2} \)
73 \( 1 + (-1.16 - 1.16i)T + 73iT^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + (-5.65 - 5.65i)T + 83iT^{2} \)
89 \( 1 + (5.06 - 5.06i)T - 89iT^{2} \)
97 \( 1 + (-5.16 + 5.16i)T - 97iT^{2} \)
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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

−11.04161602440410691815654506459, −10.40913308891471711300387201442, −9.115290878989004299585195909691, −8.872509610274487890706877977613, −7.87472866141934935504261711543, −6.86913082653819243260356392906, −5.20444884915956434525320809054, −3.89312823559299478481531099645, −3.12390734432927115711334986736, −1.45708633879871916704372554995, 1.42048428454966644722053941760, 3.06273931917052652215031047567, 4.25924257738273254236956166580, 5.91149871057404279815297651021, 6.90558507922383242984379306906, 7.66465788939106789943577139889, 8.505313875858648815574922900895, 9.310069222986089719251893786688, 10.18933536108040880680020979912, 11.27082751892949899950542195139

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