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Properties

Label 2-390-65.32-c1-0-0
Degree $2$
Conductor $390$
Sign $-0.904 - 0.427i$
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 + (−0.258 − 0.965i)3-s + (−0.499 − 0.866i)4-s + (−0.849 − 2.06i)5-s + (0.965 + 0.258i)6-s + (−1.91 + 1.10i)7-s + 0.999·8-s + (−0.866 + 0.499i)9-s + (2.21 + 0.298i)10-s + (−5.02 + 1.34i)11-s + (−0.707 + 0.707i)12-s + (−0.134 + 3.60i)13-s − 2.21i·14-s + (−1.77 + 1.35i)15-s + (−0.5 + 0.866i)16-s + (6.47 + 1.73i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.149 − 0.557i)3-s + (−0.249 − 0.433i)4-s + (−0.379 − 0.925i)5-s + (0.394 + 0.105i)6-s + (−0.725 + 0.418i)7-s + 0.353·8-s + (−0.288 + 0.166i)9-s + (0.700 + 0.0943i)10-s + (−1.51 + 0.405i)11-s + (−0.204 + 0.204i)12-s + (−0.0373 + 0.999i)13-s − 0.592i·14-s + (−0.459 + 0.350i)15-s + (−0.125 + 0.216i)16-s + (1.57 + 0.421i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.904 - 0.427i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.904 - 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0456277 + 0.203250i\)
\(L(\frac12)\) \(\approx\) \(0.0456277 + 0.203250i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (0.849 + 2.06i)T \)
13 \( 1 + (0.134 - 3.60i)T \)
good7 \( 1 + (1.91 - 1.10i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.02 - 1.34i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-6.47 - 1.73i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.46 - 5.47i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.38 - 0.371i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (5.81 + 3.35i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.26 - 3.26i)T - 31iT^{2} \)
37 \( 1 + (1.20 + 0.694i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.84 + 10.6i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.24 + 8.38i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 8.61iT - 47T^{2} \)
53 \( 1 + (6.12 - 6.12i)T - 53iT^{2} \)
59 \( 1 + (5.50 + 1.47i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.22 - 2.11i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.05 - 8.75i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.59 + 1.49i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 5.26T + 73T^{2} \)
79 \( 1 + 2.18iT - 79T^{2} \)
83 \( 1 - 13.9iT - 83T^{2} \)
89 \( 1 + (-1.36 - 5.08i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (4.69 + 8.13i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.11040371587051447150445870729, −10.59284988332310176372213274008, −9.739271053750460880307120621366, −8.778001295575677962588441300228, −7.898022097169160755951978562710, −7.25993703979160232386940603777, −5.85439456909298274919068139114, −5.32349412697459039773568803911, −3.79386970000658905814836985085, −1.85904204718376428232898123789, 0.15022099397040457587994840535, 2.92804874111419657502683630351, 3.32032212119411288340526559571, 4.89513871795782537632509678631, 6.07123262623848790413277512753, 7.47584091702209994058935420972, 8.022781663508293906172486501776, 9.504862239903874813603349011595, 10.15723499940188401862261048380, 10.82486148863047166504740376713

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