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Properties

Label 2-390-65.9-c1-0-3
Degree $2$
Conductor $390$
Sign $-0.0727 - 0.997i$
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 + i)5-s + (−0.499 − 0.866i)6-s + (−4.33 + 2.5i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (1.23 + 1.86i)10-s + (−1.5 + 2.59i)11-s − 0.999i·12-s + (2.59 + 2.5i)13-s − 5·14-s + (−1.23 − 1.86i)15-s + (−0.5 + 0.866i)16-s + (3.46 − 2i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.894 + 0.447i)5-s + (−0.204 − 0.353i)6-s + (−1.63 + 0.944i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.389 + 0.590i)10-s + (−0.452 + 0.783i)11-s − 0.288i·12-s + (0.720 + 0.693i)13-s − 1.33·14-s + (−0.318 − 0.481i)15-s + (−0.125 + 0.216i)16-s + (0.840 − 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0727 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0727 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.0727 - 0.997i$
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: \((2,\ 390,\ (\ :1/2),\ -0.0727 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03855 + 1.11702i\)
\(L(\frac12)\) \(\approx\) \(1.03855 + 1.11702i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - i)T \)
13 \( 1 + (-2.59 - 2.5i)T \)
good7 \( 1 + (4.33 - 2.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.46 + 2i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (7.79 + 4.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5 - 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.3 + 6i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.19 - 3i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6 + 10.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 16iT - 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 + 10iT - 83T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-44.5 - 77.0i)T^{2} \)
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

−11.89720684998803054768101750939, −10.61156231513847102377851983871, −9.744112882774344364413376600172, −8.969803564301572884120301305166, −7.40237288842277226922411069453, −6.46468746572719070163562776483, −6.02870345892483446219302293580, −5.01761637383860217653087170267, −3.33920240179346243036148945605, −2.25732713056652101146486863066, 0.904731430141419157061923437734, 3.04412745765232313891655648181, 3.91103772676776058047965375344, 5.40405108149311667109437889156, 6.02036717452633514760973348306, 6.87589049076241351112091511031, 8.460489531016420661099955098123, 9.684323321434553349405455017218, 10.32322412569770685873306567647, 10.78262109554102343645018433810

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