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Properties

Label 2-390-65.33-c1-0-0
Degree $2$
Conductor $390$
Sign $-0.516 + 0.856i$
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 + (0.0142 + 2.23i)5-s + (0.258 − 0.965i)6-s + (−3.50 + 2.02i)7-s + 0.999·8-s + (0.866 − 0.499i)9-s + (−1.94 − 1.10i)10-s + (−1.40 − 5.23i)11-s + (0.707 + 0.707i)12-s + (2.49 + 2.60i)13-s − 4.04i·14-s + (−0.592 − 2.15i)15-s + (−0.5 + 0.866i)16-s + (−0.0494 + 0.184i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.557 + 0.149i)3-s + (−0.249 − 0.433i)4-s + (0.00635 + 0.999i)5-s + (0.105 − 0.394i)6-s + (−1.32 + 0.764i)7-s + 0.353·8-s + (0.288 − 0.166i)9-s + (−0.614 − 0.349i)10-s + (−0.423 − 1.57i)11-s + (0.204 + 0.204i)12-s + (0.691 + 0.721i)13-s − 1.08i·14-s + (−0.152 − 0.556i)15-s + (−0.125 + 0.216i)16-s + (−0.0120 + 0.0448i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.516 + 0.856i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.516 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0465111 - 0.0823628i\)
\(L(\frac12)\) \(\approx\) \(0.0465111 - 0.0823628i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0142 - 2.23i)T \)
13 \( 1 + (-2.49 - 2.60i)T \)
good7 \( 1 + (3.50 - 2.02i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.40 + 5.23i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.0494 - 0.184i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (6.09 + 1.63i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.67 + 6.25i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.98 + 1.72i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.55 - 3.55i)T + 31iT^{2} \)
37 \( 1 + (6.84 + 3.95i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.73 - 1.53i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.93 - 1.32i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 2.11iT - 47T^{2} \)
53 \( 1 + (-0.255 - 0.255i)T + 53iT^{2} \)
59 \( 1 + (-1.13 + 4.22i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.82 - 8.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.54 - 11.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.0786 + 0.293i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 13.3T + 73T^{2} \)
79 \( 1 - 2.94iT - 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + (6.09 - 1.63i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.47 - 7.75i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.70477503418410771523276082265, −10.77735122552553131677975000511, −10.24666508106632784257166217388, −9.033656442761837251551319014044, −8.398738884937313883739312667044, −6.81693916897189059513601002975, −6.31832834210262615472588013811, −5.69239761736107186186522999116, −3.94221866235160931653590206367, −2.68139678238664817988800626851, 0.07239697235493678801645744112, 1.74224990249566103502302734002, 3.60371421503965384518385550543, 4.59044033553451419633122011943, 5.84304810993629391889088664868, 7.02250148034813936029469814787, 7.949380357217311801973592077533, 9.111672549591592238119955949553, 10.07152532338432521063739419148, 10.39071475708711266734747921404

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