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Properties

Label 2-390-65.2-c1-0-2
Degree $2$
Conductor $390$
Sign $0.956 - 0.291i$
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 + (0.965 + 0.258i)3-s + (−0.499 + 0.866i)4-s + (−2.23 + 0.0407i)5-s + (−0.258 − 0.965i)6-s + (2.13 + 1.23i)7-s + 0.999·8-s + (0.866 + 0.499i)9-s + (1.15 + 1.91i)10-s + (−1.25 + 4.66i)11-s + (−0.707 + 0.707i)12-s + (3.10 − 1.83i)13-s − 2.47i·14-s + (−2.17 − 0.539i)15-s + (−0.5 − 0.866i)16-s + (1.06 + 3.96i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.557 + 0.149i)3-s + (−0.249 + 0.433i)4-s + (−0.999 + 0.0182i)5-s + (−0.105 − 0.394i)6-s + (0.808 + 0.466i)7-s + 0.353·8-s + (0.288 + 0.166i)9-s + (0.364 + 0.605i)10-s + (−0.377 + 1.40i)11-s + (−0.204 + 0.204i)12-s + (0.860 − 0.510i)13-s − 0.660i·14-s + (−0.560 − 0.139i)15-s + (−0.125 − 0.216i)16-s + (0.257 + 0.960i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.956 - 0.291i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.956 - 0.291i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21494 + 0.181089i\)
\(L(\frac12)\) \(\approx\) \(1.21494 + 0.181089i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (2.23 - 0.0407i)T \)
13 \( 1 + (-3.10 + 1.83i)T \)
good7 \( 1 + (-2.13 - 1.23i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.25 - 4.66i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.06 - 3.96i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.34 + 0.896i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.548 - 2.04i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-6.06 + 3.50i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.98 - 2.98i)T - 31iT^{2} \)
37 \( 1 + (3.86 - 2.23i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.437 + 0.117i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.79 - 0.479i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 3.40iT - 47T^{2} \)
53 \( 1 + (-2.58 + 2.58i)T - 53iT^{2} \)
59 \( 1 + (2.64 + 9.87i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.89 + 5.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.17 - 8.96i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.51 + 13.1i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 5.15T + 73T^{2} \)
79 \( 1 - 5.85iT - 79T^{2} \)
83 \( 1 - 7.04iT - 83T^{2} \)
89 \( 1 + (14.1 + 3.77i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (9.50 - 16.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.31713616338256914040510969581, −10.47903665947719876426247932979, −9.594636851978382147337377573684, −8.358046312576531802634715602956, −8.089466059221285754361778894848, −7.02157305989701556389788183387, −5.19249975637908113684798076368, −4.18461911116431008308307726341, −3.10337178767964079766871957154, −1.65339084297721372058328235938, 0.996493970573337016821213301897, 3.17571056707543308567869723841, 4.30364955711357074967066162123, 5.50695730311496861397882674706, 6.87108396366884351913099844435, 7.68703586320861411667339568040, 8.405325616685270266770808915444, 8.992258099972683357438915695230, 10.41997104627056475987605262961, 11.21066177822366979652834156413

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