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Properties

Label 2-390-65.32-c1-0-3
Degree $2$
Conductor $390$
Sign $-0.997 + 0.0693i$
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 + (−0.743 + 2.10i)5-s + (−0.965 − 0.258i)6-s + (−0.0594 + 0.0343i)7-s + 0.999·8-s + (−0.866 + 0.499i)9-s + (−1.45 − 1.69i)10-s + (−5.23 + 1.40i)11-s + (0.707 − 0.707i)12-s + (3.36 + 1.29i)13-s − 0.0686i·14-s + (−2.22 − 0.172i)15-s + (−0.5 + 0.866i)16-s + (−3.28 − 0.880i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.149 + 0.557i)3-s + (−0.249 − 0.433i)4-s + (−0.332 + 0.943i)5-s + (−0.394 − 0.105i)6-s + (−0.0224 + 0.0129i)7-s + 0.353·8-s + (−0.288 + 0.166i)9-s + (−0.460 − 0.536i)10-s + (−1.57 + 0.422i)11-s + (0.204 − 0.204i)12-s + (0.932 + 0.360i)13-s − 0.0183i·14-s + (−0.575 − 0.0444i)15-s + (−0.125 + 0.216i)16-s + (−0.796 − 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.997 + 0.0693i$
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: \((2,\ 390,\ (\ :1/2),\ -0.997 + 0.0693i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0245664 - 0.707658i\)
\(L(\frac12)\) \(\approx\) \(0.0245664 - 0.707658i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.743 - 2.10i)T \)
13 \( 1 + (-3.36 - 1.29i)T \)
good7 \( 1 + (0.0594 - 0.0343i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.23 - 1.40i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (3.28 + 0.880i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.295 - 1.10i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.72 - 0.462i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.37 - 1.94i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.29 - 2.29i)T - 31iT^{2} \)
37 \( 1 + (3.81 + 2.20i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.47 - 5.49i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.29 + 8.55i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 9.48iT - 47T^{2} \)
53 \( 1 + (-5.48 + 5.48i)T - 53iT^{2} \)
59 \( 1 + (-5.49 - 1.47i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.87 - 6.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.519 - 0.899i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.00 + 1.34i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 - 11.5iT - 79T^{2} \)
83 \( 1 - 4.93iT - 83T^{2} \)
89 \( 1 + (-0.0552 - 0.206i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-6.94 - 12.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.39551598660950711140853688795, −10.64923449152695182417572997217, −10.08756220027972112958692277670, −8.926510837370173259828717180595, −8.048350800374333074202686434842, −7.18774198737982183353857913243, −6.18520358148244198009068680660, −5.05013279202712533612619323664, −3.84387041530146723701260378142, −2.47342796303004286161616425702, 0.49869636101257146740765032780, 2.14611697366108119158468274640, 3.52129600995861328426859048838, 4.85144679152962708441829088968, 5.96600559127009504654582883606, 7.43468909603371309921554071025, 8.359502239585832517679566195790, 8.703979143808218619590696016179, 10.02129794043710612537468794546, 10.93610454798397899481391939035

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