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Properties

Label 2-390-65.2-c1-0-6
Degree $2$
Conductor $390$
Sign $0.135 + 0.990i$
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 + (1.23 + 1.86i)5-s + (0.258 + 0.965i)6-s + (−3.19 − 1.84i)7-s + 0.999·8-s + (0.866 + 0.499i)9-s + (0.994 − 2.00i)10-s + (0.762 − 2.84i)11-s + (0.707 − 0.707i)12-s + (3.53 + 0.692i)13-s + 3.68i·14-s + (−0.713 − 2.11i)15-s + (−0.5 − 0.866i)16-s + (−1.01 − 3.79i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.557 − 0.149i)3-s + (−0.249 + 0.433i)4-s + (0.553 + 0.832i)5-s + (0.105 + 0.394i)6-s + (−1.20 − 0.696i)7-s + 0.353·8-s + (0.288 + 0.166i)9-s + (0.314 − 0.633i)10-s + (0.229 − 0.858i)11-s + (0.204 − 0.204i)12-s + (0.981 + 0.192i)13-s + 0.985i·14-s + (−0.184 − 0.547i)15-s + (−0.125 − 0.216i)16-s + (−0.246 − 0.921i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.135 + 0.990i$
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: \((2,\ 390,\ (\ :1/2),\ 0.135 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.677049 - 0.590890i\)
\(L(\frac12)\) \(\approx\) \(0.677049 - 0.590890i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 1.86i)T \)
13 \( 1 + (-3.53 - 0.692i)T \)
good7 \( 1 + (3.19 + 1.84i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.762 + 2.84i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.01 + 3.79i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-7.83 + 2.09i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.89 + 7.08i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-4.63 + 2.67i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.94 - 2.94i)T - 31iT^{2} \)
37 \( 1 + (1.96 - 1.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.80 + 0.750i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (3.65 - 0.980i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 12.3iT - 47T^{2} \)
53 \( 1 + (0.952 - 0.952i)T - 53iT^{2} \)
59 \( 1 + (-1.76 - 6.59i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.93 + 8.54i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.56 - 11.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.620 - 2.31i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 - 8.67T + 73T^{2} \)
79 \( 1 - 10.8iT - 79T^{2} \)
83 \( 1 - 10.4iT - 83T^{2} \)
89 \( 1 + (9.57 + 2.56i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.60 - 2.77i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.08499560387176249083573672290, −10.23470499216635705529946270225, −9.616740271674657397341221893604, −8.551828421593362289640842470657, −6.96656808629006890341807566899, −6.65428684983251806371597826874, −5.36556043742683886599383327406, −3.68226243335880679377206198679, −2.82553905556557931996732051414, −0.818465504099800565622918541777, 1.42909005264919414003079172399, 3.56509046592107403546296658363, 5.08369210634988255099255607581, 5.83901932711433287431263093003, 6.55846052087135339772645022536, 7.80405499735355663244736398833, 9.038981155618396056276862782933, 9.524877337578290659875060286638, 10.26209221613020173890472746450, 11.58247556040426184457486863474

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