↑

Properties

Label 2-504-21.5-c3-0-8
Degree $2$
Conductor $504$
Sign $-0.0190 - 0.999i$
Analytic cond. $29.7369$
Root an. cond. $5.45316$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−7.29 + 12.6i)5-s + (18.4 − 0.998i)7-s + (38.7 − 22.3i)11-s + 76.8i·13-s + (−58.3 − 101. i)17-s + (83.5 + 48.2i)19-s + (−6.88 − 3.97i)23-s + (−43.9 − 76.1i)25-s + 86.8i·29-s + (216. − 124. i)31-s + (−122. + 240. i)35-s + (−160. + 277. i)37-s + 231.·41-s − 413.·43-s + (−235. + 407. i)47-s + ⋯
L(s)  = 1  + (−0.652 + 1.13i)5-s + (0.998 − 0.0539i)7-s + (1.06 − 0.612i)11-s + 1.63i·13-s + (−0.832 − 1.44i)17-s + (1.00 + 0.582i)19-s + (−0.0624 − 0.0360i)23-s + (−0.351 − 0.608i)25-s + 0.556i·29-s + (1.25 − 0.723i)31-s + (−0.590 + 1.16i)35-s + (−0.711 + 1.23i)37-s + 0.881·41-s − 1.46·43-s + (−0.730 + 1.26i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0190 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0190 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.0190 - 0.999i$
Analytic conductor: \(29.7369\)
Root analytic conductor: \(5.45316\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :3/2),\ -0.0190 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.838196605\)
\(L(\frac12)\) \(\approx\) \(1.838196605\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 + (-18.4 + 0.998i)T \)
good5 \( 1 + (7.29 - 12.6i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-38.7 + 22.3i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 76.8iT - 2.19e3T^{2} \)
17 \( 1 + (58.3 + 101. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-83.5 - 48.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (6.88 + 3.97i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 86.8iT - 2.43e4T^{2} \)
31 \( 1 + (-216. + 124. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (160. - 277. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 231.T + 6.89e4T^{2} \)
43 \( 1 + 413.T + 7.95e4T^{2} \)
47 \( 1 + (235. - 407. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (600. - 346. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (143. + 249. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-740. - 427. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-240. - 416. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 930. iT - 3.57e5T^{2} \)
73 \( 1 + (98.4 - 56.8i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-111. + 192. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 692.T + 5.71e5T^{2} \)
89 \( 1 + (-258. + 448. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 807. iT - 9.12e5T^{2} \)
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

−11.10618406864615529602786759979, −9.815648602773632013451425888032, −8.955191184117622098827051316489, −7.945306174698130787169847883229, −6.99638281354922962189022225434, −6.44199074314580001260335052551, −4.87292209534641988184005105523, −3.96926546164837153881724359454, −2.81640823402894943026973511698, −1.34826482371189837844464148733, 0.63688570501979133371432839032, 1.77273744854766971883636674423, 3.59905673006978955472055518870, 4.62008118743034792354791557531, 5.30345281321040174936392506702, 6.63646984682970245147594121927, 7.934040783747911062017867329452, 8.313447779906462203978711501501, 9.225747302051458255443631649760, 10.33557697820211338612345203894

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