Properties

Label 2-390-195.44-c1-0-4
Degree $2$
Conductor $390$
Sign $0.245 - 0.969i$
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 + (−1.46 − 0.926i)3-s − 1.00i·4-s + (−0.989 + 2.00i)5-s + (−1.68 + 0.379i)6-s + (−1.27 + 1.27i)7-s + (−0.707 − 0.707i)8-s + (1.28 + 2.71i)9-s + (0.717 + 2.11i)10-s + (−2.42 + 2.42i)11-s + (−0.926 + 1.46i)12-s + (−1.08 + 3.43i)13-s + 1.80i·14-s + (3.30 − 2.01i)15-s − 1.00·16-s + 1.22i·17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.844 − 0.534i)3-s − 0.500i·4-s + (−0.442 + 0.896i)5-s + (−0.689 + 0.155i)6-s + (−0.483 + 0.483i)7-s + (−0.250 − 0.250i)8-s + (0.427 + 0.903i)9-s + (0.226 + 0.669i)10-s + (−0.732 + 0.732i)11-s + (−0.267 + 0.422i)12-s + (−0.300 + 0.953i)13-s + 0.483i·14-s + (0.853 − 0.520i)15-s − 0.250·16-s + 0.297i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.245 - 0.969i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.245 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.550437 + 0.428214i\)
\(L(\frac12)\) \(\approx\) \(0.550437 + 0.428214i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.46 + 0.926i)T \)
5 \( 1 + (0.989 - 2.00i)T \)
13 \( 1 + (1.08 - 3.43i)T \)
good7 \( 1 + (1.27 - 1.27i)T - 7iT^{2} \)
11 \( 1 + (2.42 - 2.42i)T - 11iT^{2} \)
17 \( 1 - 1.22iT - 17T^{2} \)
19 \( 1 + (-3.93 + 3.93i)T - 19iT^{2} \)
23 \( 1 + 0.355iT - 23T^{2} \)
29 \( 1 - 9.56iT - 29T^{2} \)
31 \( 1 + (7.14 - 7.14i)T - 31iT^{2} \)
37 \( 1 + (2.19 - 2.19i)T - 37iT^{2} \)
41 \( 1 + (6.56 + 6.56i)T + 41iT^{2} \)
43 \( 1 - 8.65T + 43T^{2} \)
47 \( 1 + (6.46 + 6.46i)T + 47iT^{2} \)
53 \( 1 - 7.42T + 53T^{2} \)
59 \( 1 + (-0.905 + 0.905i)T - 59iT^{2} \)
61 \( 1 - 2.59T + 61T^{2} \)
67 \( 1 + (-1.26 - 1.26i)T + 67iT^{2} \)
71 \( 1 + (6.55 + 6.55i)T + 71iT^{2} \)
73 \( 1 + (9.28 - 9.28i)T - 73iT^{2} \)
79 \( 1 + 6.20T + 79T^{2} \)
83 \( 1 + (0.300 - 0.300i)T - 83iT^{2} \)
89 \( 1 + (-10.8 + 10.8i)T - 89iT^{2} \)
97 \( 1 + (-1.85 - 1.85i)T + 97iT^{2} \)
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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

−11.62493457042352681130489965535, −10.76927555857367825406969163830, −10.11946850643177018522589572695, −8.879274175245964469021181399065, −7.13872304187581665450073424979, −6.99744619500351405018514547733, −5.62888316557726691044264150907, −4.73867734818368873597690497352, −3.24576540287880598963892207885, −2.00851366244285041953739739433, 0.42186751972062168658409564878, 3.35312735903724669819490126113, 4.29907984219438425741656147247, 5.43119657084482424995209358664, 5.90274378254854165372405097792, 7.39856893485860443033029154839, 8.095712499382625447806803216874, 9.417628433404673334358554041698, 10.20094347655375383970987251098, 11.32275674513504416313491800249

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