↑

Properties

Label 2-504-63.4-c1-0-21
Degree $2$
Conductor $504$
Sign $-0.853 - 0.521i$
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  + (−1.04 − 1.38i)3-s − 0.0619·5-s + (−1.63 − 2.07i)7-s + (−0.828 + 2.88i)9-s − 3.18·11-s + (−0.252 + 0.437i)13-s + (0.0645 + 0.0857i)15-s + (−0.554 + 0.960i)17-s + (0.933 + 1.61i)19-s + (−1.16 + 4.43i)21-s − 6.20·23-s − 4.99·25-s + (4.85 − 1.85i)27-s + (2.39 + 4.15i)29-s + (1.26 + 2.19i)31-s + ⋯
L(s)  = 1  + (−0.601 − 0.798i)3-s − 0.0277·5-s + (−0.618 − 0.785i)7-s + (−0.276 + 0.961i)9-s − 0.958·11-s + (−0.0700 + 0.121i)13-s + (0.0166 + 0.0221i)15-s + (−0.134 + 0.233i)17-s + (0.214 + 0.370i)19-s + (−0.255 + 0.966i)21-s − 1.29·23-s − 0.999·25-s + (0.933 − 0.357i)27-s + (0.445 + 0.770i)29-s + (0.227 + 0.394i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.853 - 0.521i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.853 - 0.521i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0305179 + 0.108444i\)
\(L(\frac12)\) \(\approx\) \(0.0305179 + 0.108444i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.04 + 1.38i)T \)
7 \( 1 + (1.63 + 2.07i)T \)
good5 \( 1 + 0.0619T + 5T^{2} \)
11 \( 1 + 3.18T + 11T^{2} \)
13 \( 1 + (0.252 - 0.437i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.554 - 0.960i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.933 - 1.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.20T + 23T^{2} \)
29 \( 1 + (-2.39 - 4.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.26 - 2.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.26 + 7.38i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.94 - 8.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.95 + 6.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.29 - 5.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.58 - 2.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.50 + 7.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.94 + 12.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.66 + 2.88i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.25T + 71T^{2} \)
73 \( 1 + (-2.07 + 3.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.48 + 2.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.17 - 3.76i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.30 + 7.44i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.27 + 5.67i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.44886870958490888858031545426, −9.786054068284845181187238161574, −8.291636952179889327579248266946, −7.61745162224314768961449254733, −6.70682943442101941329267473511, −5.87304303609717554391468694159, −4.78706272033078112926141454985, −3.40036707908103213339346621024, −1.86694089124005907912493350206, −0.06760020064369572246790728837, 2.53167665062600038268030748476, 3.73330259203713821202953283256, 4.97418448480802350585478024563, 5.75801531765672803652117889362, 6.61429872839689419818082211801, 7.978667459731989141719670116011, 8.910845954375914723867339008055, 9.984812242873323205593241195895, 10.23295029040486172516781855616, 11.65189443723025222733212939802

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