↑

Properties

Label 2-504-63.16-c1-0-14
Degree $2$
Conductor $504$
Sign $0.985 - 0.171i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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  + (1.70 − 0.327i)3-s + 0.340·5-s + (1.09 + 2.40i)7-s + (2.78 − 1.11i)9-s − 0.671·11-s + (1.62 + 2.81i)13-s + (0.578 − 0.111i)15-s + (−1.10 − 1.90i)17-s + (0.242 − 0.419i)19-s + (2.65 + 3.73i)21-s + 4.18·23-s − 4.88·25-s + (4.37 − 2.80i)27-s + (0.478 − 0.829i)29-s + (−1.04 + 1.80i)31-s + ⋯
L(s)  = 1  + (0.982 − 0.188i)3-s + 0.152·5-s + (0.414 + 0.909i)7-s + (0.928 − 0.370i)9-s − 0.202·11-s + (0.450 + 0.779i)13-s + (0.149 − 0.0287i)15-s + (−0.266 − 0.462i)17-s + (0.0555 − 0.0961i)19-s + (0.579 + 0.815i)21-s + 0.873·23-s − 0.976·25-s + (0.842 − 0.539i)27-s + (0.0889 − 0.154i)29-s + (−0.187 + 0.323i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.985 - 0.171i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.985 - 0.171i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11485 + 0.183227i\)
\(L(\frac12)\) \(\approx\) \(2.11485 + 0.183227i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.70 + 0.327i)T \)
7 \( 1 + (-1.09 - 2.40i)T \)
good5 \( 1 - 0.340T + 5T^{2} \)
11 \( 1 + 0.671T + 11T^{2} \)
13 \( 1 + (-1.62 - 2.81i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.10 + 1.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.242 + 0.419i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.18T + 23T^{2} \)
29 \( 1 + (-0.478 + 0.829i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.04 - 1.80i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.81 + 8.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.90 + 6.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.66 - 6.34i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.34 - 2.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.12 + 10.6i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.47 + 4.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.76 - 3.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.16 - 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.57T + 71T^{2} \)
73 \( 1 + (3.71 + 6.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.00 - 8.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.47 + 4.28i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.52 - 14.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.23 + 7.33i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.02915002647898284241513502324, −9.746445188264057498841072821336, −9.086587369263811927527365742912, −8.400742308656895771124719041275, −7.44438916740888672101436133262, −6.45618454519795433284488056248, −5.25246427255664355087305927228, −4.07319074402735822114784426033, −2.78020039898462728630372981415, −1.75679084278398295334929027753, 1.46951063711782068416245938597, 2.98501182277320365681641407763, 4.00266730084547284570454713231, 5.01381090214505543076088814433, 6.41305573118515153948199355865, 7.56500043841907729360028980438, 8.120189016157575029481768519809, 9.063606478264666707646370273150, 10.10974049043077913642021086186, 10.60892843853888758367459489618

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