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Properties

Label 2-390-65.63-c1-0-1
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: $\chi_{390} (193, \cdot )$
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

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

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