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Properties

Label 2-390-65.9-c1-0-0
Degree $2$
Conductor $390$
Sign $-0.965 - 0.260i$
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  + (0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−2.23 − 0.133i)5-s + (−0.499 − 0.866i)6-s + (−2.36 + 1.36i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−1.86 − 1.23i)10-s + (−0.366 + 0.633i)11-s − 0.999i·12-s + (−3.59 + 0.232i)13-s − 2.73·14-s + (1.86 + 1.23i)15-s + (−0.5 + 0.866i)16-s + (−3.86 + 2.23i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.998 − 0.0599i)5-s + (−0.204 − 0.353i)6-s + (−0.894 + 0.516i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.590 − 0.389i)10-s + (−0.110 + 0.191i)11-s − 0.288i·12-s + (−0.997 + 0.0643i)13-s − 0.730·14-s + (0.481 + 0.318i)15-s + (−0.125 + 0.216i)16-s + (−0.937 + 0.541i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.965 - 0.260i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.965 - 0.260i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0603893 + 0.455255i\)
\(L(\frac12)\) \(\approx\) \(0.0603893 + 0.455255i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (2.23 + 0.133i)T \)
13 \( 1 + (3.59 - 0.232i)T \)
good7 \( 1 + (2.36 - 1.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.366 - 0.633i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.86 - 2.23i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.633 - 1.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.36 + 3.09i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.23 - 5.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-6.86 - 3.96i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.59 + 7.96i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.83 - 1.63i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.66iT - 47T^{2} \)
53 \( 1 - 7.73iT - 53T^{2} \)
59 \( 1 + (6.19 + 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.06 - 8.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.63 - 3.83i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.633 + 1.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.66iT - 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 - 5.26iT - 83T^{2} \)
89 \( 1 + (2.46 - 4.26i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.66 + 5i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.05323726963825943302583146564, −11.08608934341631854888372000997, −10.02081108210021314527398404856, −8.791246470061368743734866507758, −7.78052009526588697359879109846, −6.89550236291627600852509452525, −6.08685642104320815327997419468, −4.90146886620983745651467393911, −3.91309089736109103429578290090, −2.52814888143233867512910702686, 0.24802731135425498525498585383, 2.76935609844392191654982738398, 3.96186596463694873454989311512, 4.69632397223621869030439562645, 6.02405583235058251375757870683, 6.98525166931229021444036387405, 7.88367739601194536375650332229, 9.458919610054597902374819440028, 10.06988608821735962959870482863, 11.23161830153075206086401180824

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