Properties

Label 2-390-65.32-c1-0-4
Degree $2$
Conductor $390$
Sign $0.878 - 0.478i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.258 + 0.965i)3-s + (−0.499 − 0.866i)4-s + (0.662 + 2.13i)5-s + (0.965 + 0.258i)6-s + (−1.12 + 0.648i)7-s − 0.999·8-s + (−0.866 + 0.499i)9-s + (2.18 + 0.493i)10-s + (2.28 − 0.611i)11-s + (0.707 − 0.707i)12-s + (3.01 + 1.97i)13-s + 1.29i·14-s + (−1.89 + 1.19i)15-s + (−0.5 + 0.866i)16-s + (6.70 + 1.79i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.149 + 0.557i)3-s + (−0.249 − 0.433i)4-s + (0.296 + 0.955i)5-s + (0.394 + 0.105i)6-s + (−0.424 + 0.245i)7-s − 0.353·8-s + (−0.288 + 0.166i)9-s + (0.689 + 0.156i)10-s + (0.688 − 0.184i)11-s + (0.204 − 0.204i)12-s + (0.836 + 0.547i)13-s + 0.346i·14-s + (−0.488 + 0.308i)15-s + (−0.125 + 0.216i)16-s + (1.62 + 0.435i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.878 - 0.478i$
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: \((2,\ 390,\ (\ :1/2),\ 0.878 - 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65850 + 0.422073i\)
\(L(\frac12)\) \(\approx\) \(1.65850 + 0.422073i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.662 - 2.13i)T \)
13 \( 1 + (-3.01 - 1.97i)T \)
good7 \( 1 + (1.12 - 0.648i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.28 + 0.611i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-6.70 - 1.79i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.22 - 4.55i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.91 - 1.31i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (5.49 + 3.16i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.24 + 6.24i)T - 31iT^{2} \)
37 \( 1 + (-2.80 - 1.61i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.19 + 8.17i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.208 - 0.778i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 + (-8.79 + 8.79i)T - 53iT^{2} \)
59 \( 1 + (5.81 + 1.55i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.10 - 3.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.331 - 0.573i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.2 + 3.28i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 5.04T + 73T^{2} \)
79 \( 1 + 13.2iT - 79T^{2} \)
83 \( 1 - 2.04iT - 83T^{2} \)
89 \( 1 + (1.16 + 4.33i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.99 - 5.19i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.52749103724893655872488185428, −10.19855153213344498812363697790, −10.03192125795788412613787133608, −8.906082378322340650630440599437, −7.73321788627487645281225441983, −6.19905264593334821443041152943, −5.78328541841620282204049039949, −3.94424859528776669216727299284, −3.44190583025812626139594017900, −1.94871026785869114948370127748, 1.14939392718124911732584926190, 3.14288155315172428723389911186, 4.42016118334997331400189400455, 5.61334894019504208851119827728, 6.38694474725883656414672807471, 7.48776701108105086154792489011, 8.363948176877556987539718713393, 9.191592582296750835962361993890, 10.11945621135322803674516950657, 11.57457686971706080510764037187

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