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Properties

Label 2-390-65.32-c1-0-5
Degree $2$
Conductor $390$
Sign $0.992 + 0.122i$
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  + (−0.5 + 0.866i)2-s + (−0.258 − 0.965i)3-s + (−0.499 − 0.866i)4-s + (−2.04 + 0.900i)5-s + (0.965 + 0.258i)6-s + (−0.0517 + 0.0298i)7-s + 0.999·8-s + (−0.866 + 0.499i)9-s + (0.242 − 2.22i)10-s + (2.65 − 0.710i)11-s + (−0.707 + 0.707i)12-s + (3.48 − 0.914i)13-s − 0.0597i·14-s + (1.39 + 1.74i)15-s + (−0.5 + 0.866i)16-s + (4.56 + 1.22i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.149 − 0.557i)3-s + (−0.249 − 0.433i)4-s + (−0.915 + 0.402i)5-s + (0.394 + 0.105i)6-s + (−0.0195 + 0.0112i)7-s + 0.353·8-s + (−0.288 + 0.166i)9-s + (0.0768 − 0.702i)10-s + (0.799 − 0.214i)11-s + (−0.204 + 0.204i)12-s + (0.967 − 0.253i)13-s − 0.0159i·14-s + (0.361 + 0.450i)15-s + (−0.125 + 0.216i)16-s + (1.10 + 0.296i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.992 + 0.122i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.992 + 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.955178 - 0.0587961i\)
\(L(\frac12)\) \(\approx\) \(0.955178 - 0.0587961i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (2.04 - 0.900i)T \)
13 \( 1 + (-3.48 + 0.914i)T \)
good7 \( 1 + (0.0517 - 0.0298i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.65 + 0.710i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-4.56 - 1.22i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.19 + 8.18i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.12 - 0.837i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.06 - 1.76i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.45 + 5.45i)T - 31iT^{2} \)
37 \( 1 + (-0.388 - 0.224i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.07 + 7.76i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.05 - 3.94i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 8.10iT - 47T^{2} \)
53 \( 1 + (3.62 - 3.62i)T - 53iT^{2} \)
59 \( 1 + (-8.24 - 2.20i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-5.07 - 8.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.90 + 8.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.76 + 0.474i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 + 12.6iT - 79T^{2} \)
83 \( 1 + 0.810iT - 83T^{2} \)
89 \( 1 + (3.49 + 13.0i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-5.32 - 9.22i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.37670506828381699900101741509, −10.45860647816374513925855072510, −9.224839190998380644218602337976, −8.313555348622166588106340365608, −7.56031413007557770899391415713, −6.67667291179975504355764362544, −5.85420191355352646092720981304, −4.41298111653060237479083778601, −3.10659197252863523158096877221, −0.938514615939891520462951322715, 1.26706503636306099501026276214, 3.43949560438389721544375968575, 4.01669404364223888938902392848, 5.28916542088448869644078246006, 6.64491450302844969496425604088, 8.061794578330488525386066827188, 8.509437115015443090972457624380, 9.755621888750466573744585043717, 10.27036108011493960385545446353, 11.62991809592368619317394152094

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