↑

Properties

Label 2-504-9.7-c1-0-6
Degree $2$
Conductor $504$
Sign $0.453 - 0.891i$
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  + (1.71 + 0.255i)3-s + (1.81 + 3.13i)5-s + (−0.5 + 0.866i)7-s + (2.86 + 0.875i)9-s + (−1.95 + 3.39i)11-s + (−2.53 − 4.39i)13-s + (2.30 + 5.83i)15-s + 1.03·17-s − 2.50·19-s + (−1.07 + 1.35i)21-s + (−2.47 − 4.29i)23-s + (−4.06 + 7.04i)25-s + (4.69 + 2.23i)27-s + (4.60 − 7.97i)29-s + (0.422 + 0.731i)31-s + ⋯
L(s)  = 1  + (0.989 + 0.147i)3-s + (0.810 + 1.40i)5-s + (−0.188 + 0.327i)7-s + (0.956 + 0.291i)9-s + (−0.590 + 1.02i)11-s + (−0.703 − 1.21i)13-s + (0.594 + 1.50i)15-s + 0.250·17-s − 0.575·19-s + (−0.235 + 0.295i)21-s + (−0.516 − 0.895i)23-s + (−0.813 + 1.40i)25-s + (0.902 + 0.429i)27-s + (0.854 − 1.48i)29-s + (0.0758 + 0.131i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.453 - 0.891i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.453 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79642 + 1.10161i\)
\(L(\frac12)\) \(\approx\) \(1.79642 + 1.10161i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.255i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-1.81 - 3.13i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.95 - 3.39i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.53 + 4.39i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.03T + 17T^{2} \)
19 \( 1 + 2.50T + 19T^{2} \)
23 \( 1 + (2.47 + 4.29i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.60 + 7.97i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.422 - 0.731i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.84T + 37T^{2} \)
41 \( 1 + (-2.07 - 3.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.20 + 3.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.93 + 6.82i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + (-5.60 - 9.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.208 - 0.360i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.02 + 8.70i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.05T + 71T^{2} \)
73 \( 1 - 7.20T + 73T^{2} \)
79 \( 1 + (7.56 - 13.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.932 - 1.61i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.669T + 89T^{2} \)
97 \( 1 + (7.63 - 13.2i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.49094648604363466228478776539, −10.23820030752057725165979977891, −9.582856156847474108001004640999, −8.293113447561071815721148741739, −7.48757279301545172988766194600, −6.64173211573827105735709802137, −5.53658973099765392535245366604, −4.16846536921387515523386264965, −2.56514680581698784611717326997, −2.51426253395555334587232469480, 1.26701256605657189366694556429, 2.52191965799985794206669360522, 3.98856523118563174658757130706, 4.97821546154016227296808287790, 6.08943426314538014575600524448, 7.28971298476458171545101849691, 8.310110716658288495040708551606, 8.992511071776569249685907941107, 9.596397470857730828802868812124, 10.49212814633730494745871373855

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