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Properties

Label 2-390-65.29-c1-0-8
Degree $2$
Conductor $390$
Sign $0.0854 + 0.996i$
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 + (1 − 2i)5-s + (0.499 − 0.866i)6-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (0.133 + 2.23i)10-s + (−1.5 − 2.59i)11-s + 0.999i·12-s + (−3.46 + i)13-s + (0.133 + 2.23i)15-s + (−0.5 − 0.866i)16-s + (−5.19 − 3i)17-s + 0.999i·18-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (0.447 − 0.894i)5-s + (0.204 − 0.353i)6-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.0423 + 0.705i)10-s + (−0.452 − 0.783i)11-s + 0.288i·12-s + (−0.960 + 0.277i)13-s + (0.0345 + 0.576i)15-s + (−0.125 − 0.216i)16-s + (−1.26 − 0.727i)17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0854 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0854 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.0854 + 0.996i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.0854 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.467213 - 0.428840i\)
\(L(\frac12)\) \(\approx\) \(0.467213 - 0.428840i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2i)T \)
13 \( 1 + (3.46 - i)T \)
good7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (5.19 + 3i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.79 + 4.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-11.2 - 6.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 11iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + (5.5 - 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.46 - 2i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + (2 + 3.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.46 + 2i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.01892894783959825082538198597, −10.04416522620046361571828408727, −9.195879376574130345367794864343, −8.580899765692809243872479005644, −7.33746365557553125324320942697, −6.32044922050739498007105159039, −5.27957983684157521839533100709, −4.52530387999619851243271388278, −2.44295096399459258998585735692, −0.52346946873224122951136894530, 1.86049703780094644396014637768, 2.99300254036233513121987740804, 4.70081728248280564190534106947, 5.95030175853905733310980143311, 7.11840521164862702860675046279, 7.49469052842598502006210800059, 9.027108011023074512545799207715, 9.799179327633031479555387741860, 10.90162689199842536304503678166, 11.01513750249578376027771424216

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