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Properties

Label 2-390-65.32-c1-0-13
Degree $2$
Conductor $390$
Sign $-0.990 - 0.134i$
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 + (−1.66 − 1.48i)5-s + (−0.965 − 0.258i)6-s + (−0.568 + 0.328i)7-s − 0.999·8-s + (−0.866 + 0.499i)9-s + (−2.12 + 0.700i)10-s + (−1.26 + 0.337i)11-s + (−0.707 + 0.707i)12-s + (0.656 − 3.54i)13-s + 0.656i·14-s + (−1.00 + 1.99i)15-s + (−0.5 + 0.866i)16-s + (−5.49 − 1.47i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.149 − 0.557i)3-s + (−0.249 − 0.433i)4-s + (−0.746 − 0.665i)5-s + (−0.394 − 0.105i)6-s + (−0.214 + 0.124i)7-s − 0.353·8-s + (−0.288 + 0.166i)9-s + (−0.671 + 0.221i)10-s + (−0.379 + 0.101i)11-s + (−0.204 + 0.204i)12-s + (0.181 − 0.983i)13-s + 0.175i·14-s + (−0.259 + 0.515i)15-s + (−0.125 + 0.216i)16-s + (−1.33 − 0.357i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.990 - 0.134i$
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: \((2,\ 390,\ (\ :1/2),\ -0.990 - 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0571605 + 0.845871i\)
\(L(\frac12)\) \(\approx\) \(0.0571605 + 0.845871i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 + 1.48i)T \)
13 \( 1 + (-0.656 + 3.54i)T \)
good7 \( 1 + (0.568 - 0.328i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.26 - 0.337i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (5.49 + 1.47i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.425 - 1.58i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.208 - 0.0557i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.0656 + 0.0378i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.13 + 5.13i)T - 31iT^{2} \)
37 \( 1 + (-0.800 - 0.462i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.01 + 7.51i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.45 + 9.17i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 7.92iT - 47T^{2} \)
53 \( 1 + (-2.58 + 2.58i)T - 53iT^{2} \)
59 \( 1 + (-4.72 - 1.26i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.60 - 2.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.64 + 11.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.94 + 1.05i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 3.44T + 73T^{2} \)
79 \( 1 - 11.0iT - 79T^{2} \)
83 \( 1 + 2.20iT - 83T^{2} \)
89 \( 1 + (1.59 + 5.96i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.83 + 11.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.07789795883093695526062556431, −10.12924914767937645407499635402, −8.912073738131605407881406125158, −8.154895602719929204938000985761, −7.08819515887611508804596684737, −5.82533372010575405339610743541, −4.85270365106916166045955387945, −3.69442198380114761407941321833, −2.30349101180065009240259613358, −0.49642932282764365294912243885, 2.82011427053011145586454955261, 4.05068852228857794179540796669, 4.76921074322712894517770856078, 6.30306904798259079308967220324, 6.85019972605158896588955654737, 8.047382525034247936814033985474, 8.891230687398606297118563598209, 10.00024388141316198150287184274, 11.06507594645875285463464259600, 11.57004617510214125622993919447

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