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Properties

Label 2-390-65.9-c1-0-11
Degree $2$
Conductor $390$
Sign $0.848 - 0.529i$
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 + (1.86 − 1.23i)10-s + (−1.5 + 2.59i)11-s + 0.999i·12-s + (3.46 + i)13-s + (1.86 − 1.23i)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 + (0.590 − 0.389i)10-s + (−0.452 + 0.783i)11-s + 0.288i·12-s + (0.960 + 0.277i)13-s + (0.481 − 0.318i)15-s + (−0.125 + 0.216i)16-s + (1.26 − 0.727i)17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.848 - 0.529i$
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: \((2,\ 390,\ (\ :1/2),\ 0.848 - 0.529i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27532 + 0.651697i\)
\(L(\frac12)\) \(\approx\) \(2.27532 + 0.651697i\)
\(L(\frac{3}{2})\) 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

−11.64193328048084738810470524685, −10.28024167203272291696145749138, −9.572092790952866251983010274195, −8.487071435563782585262446227217, −7.81252339959170431792992238353, −6.52115377836540307523463450004, −5.40095545594166892986445322971, −4.59600379550292898199093262875, −3.43246629461652458233756253495, −1.87860998897190478289866382792, 1.73342209136597312468964981183, 3.09661721052871470526448639099, 3.81633843893756395159065008809, 5.73619607287467534759947723992, 6.11866823091013366769452632513, 7.52968644524750381490817055345, 8.308745854675354611959574869291, 9.710860494368280733044306253026, 10.36871033270837601162676465150, 11.26852251774804443977652924519

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