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Properties

Label 2-390-39.8-c1-0-2
Degree $2$
Conductor $390$
Sign $-0.789 - 0.614i$
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 + (−1.70 − 0.292i)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 + (−1.70 − 0.292i)15-s − 1.00·16-s − 4.24·17-s + (−1.29 − 2.70i)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 + (−0.696 − 0.119i)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 + (−0.440 − 0.0756i)15-s − 0.250·16-s − 1.02·17-s + (−0.304 − 0.638i)18-s + (1.37 + 1.37i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.789 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.789 - 0.614i$
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: \((2,\ 390,\ (\ :1/2),\ -0.789 - 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.338504 + 0.986146i\)
\(L(\frac12)\) \(\approx\) \(0.338504 + 0.986146i\)
\(L(\frac{3}{2})\) 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

−11.40834894612124093803376400960, −10.55579055220919664340553580919, −9.554308198601143989746839679692, −9.177415289507952277780629733266, −7.88352141201085335872112352848, −7.31775538686263285416472727303, −5.99129514401020568825503883306, −4.74855835723566852369912650104, −3.75476446524374393398850826685, −2.19553677753245923300945769963, 0.76639050194924002143569331056, 2.39367046731321102593747668067, 3.46325832119581224401560127071, 4.93931939876387097826589277771, 6.52395162277621688711317129857, 7.42630613824105012773309796664, 8.232713470840360096108154024278, 9.111435551954309844953359546284, 9.776002181501443931890110735056, 11.25905466866640928362952689025

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