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Properties

Label 2-390-65.4-c1-0-8
Degree $2$
Conductor $390$
Sign $0.127 + 0.991i$
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 + (−0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (1.26 + 1.84i)5-s + (0.866 − 0.499i)6-s + (−2.17 − 3.76i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (−2.22 + 0.178i)10-s + (−2.04 − 1.17i)11-s + 0.999i·12-s + (−3.18 − 1.69i)13-s + 4.34·14-s + (−0.178 − 2.22i)15-s + (−0.5 + 0.866i)16-s + (2.60 − 1.50i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.567 + 0.823i)5-s + (0.353 − 0.204i)6-s + (−0.821 − 1.42i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.704 + 0.0563i)10-s + (−0.615 − 0.355i)11-s + 0.288i·12-s + (−0.883 − 0.469i)13-s + 1.16·14-s + (−0.0459 − 0.575i)15-s + (−0.125 + 0.216i)16-s + (0.631 − 0.364i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.127 + 0.991i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.127 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.439328 - 0.386460i\)
\(L(\frac12)\) \(\approx\) \(0.439328 - 0.386460i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-1.26 - 1.84i)T \)
13 \( 1 + (3.18 + 1.69i)T \)
good7 \( 1 + (2.17 + 3.76i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.04 + 1.17i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.60 + 1.50i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.585 + 0.338i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.58 + 3.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.82 + 8.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.11iT - 31T^{2} \)
37 \( 1 + (-3.74 + 6.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.60 + 1.50i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.91 - 3.41i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.61T + 47T^{2} \)
53 \( 1 - 9.43iT - 53T^{2} \)
59 \( 1 + (4.56 - 2.63i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.15 - 3.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.91 - 5.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.52 + 1.45i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.67T + 73T^{2} \)
79 \( 1 + 3.74T + 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 + (-4.15 - 2.39i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.17 + 14.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.71109617504674970876539528052, −10.15916638352330815746725625271, −9.640248955192947349365718590260, −7.83861970910049730257213569683, −7.38287668789310893529104093116, −6.38457298598746634983671582287, −5.70179462208914656363869880362, −4.23227505408380106987220355828, −2.66602949602715065844138439340, −0.44522657724787774130968135510, 1.84843441168044560255644499147, 3.15836002058783795694549406915, 4.84270354482072669378724094539, 5.51882273774921255350172985782, 6.65186088155334179099411705598, 8.179805733743700804553915571278, 9.044013287606965401288201406025, 9.819054857157749670881540932311, 10.28445355177561176020028081078, 11.81362523027541411414883973770

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