Properties

Label 2-504-21.5-c3-0-17
Degree $2$
Conductor $504$
Sign $-0.185 + 0.982i$
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  + (1.35 − 2.35i)5-s + (−8.74 + 16.3i)7-s + (−8.39 + 4.84i)11-s − 67.7i·13-s + (50.1 + 86.7i)17-s + (−59.6 − 34.4i)19-s + (−126. − 73.3i)23-s + (58.8 + 101. i)25-s − 284. i·29-s + (197. − 113. i)31-s + (26.5 + 42.7i)35-s + (150. − 261. i)37-s − 232.·41-s + 173.·43-s + (−191. + 331. i)47-s + ⋯
L(s)  = 1  + (0.121 − 0.210i)5-s + (−0.471 + 0.881i)7-s + (−0.229 + 0.132i)11-s − 1.44i·13-s + (0.714 + 1.23i)17-s + (−0.720 − 0.416i)19-s + (−1.15 − 0.664i)23-s + (0.470 + 0.814i)25-s − 1.82i·29-s + (1.14 − 0.659i)31-s + (0.128 + 0.206i)35-s + (0.669 − 1.16i)37-s − 0.885·41-s + 0.613·43-s + (−0.594 + 1.02i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.185 + 0.982i$
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: \((2,\ 504,\ (\ :3/2),\ -0.185 + 0.982i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.085903419\)
\(L(\frac12)\) \(\approx\) \(1.085903419\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.74 - 16.3i)T \)
good5 \( 1 + (-1.35 + 2.35i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (8.39 - 4.84i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 67.7iT - 2.19e3T^{2} \)
17 \( 1 + (-50.1 - 86.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (59.6 + 34.4i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (126. + 73.3i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 284. iT - 2.43e4T^{2} \)
31 \( 1 + (-197. + 113. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-150. + 261. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 232.T + 6.89e4T^{2} \)
43 \( 1 - 173.T + 7.95e4T^{2} \)
47 \( 1 + (191. - 331. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (22.2 - 12.8i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (371. + 644. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (343. + 198. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (293. + 508. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 501. iT - 3.57e5T^{2} \)
73 \( 1 + (-616. + 356. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-466. + 807. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 837.T + 5.71e5T^{2} \)
89 \( 1 + (-442. + 766. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.17e3iT - 9.12e5T^{2} \)
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

−10.17408826757553859896325094446, −9.481175177568772381733201493790, −8.304518887031029633761270421894, −7.85096814773934549279289337195, −6.20173981149606987172745470432, −5.82182086358739799371118229337, −4.53525491174162834025738466902, −3.21864993324701611502762055444, −2.11953912594285262243678143051, −0.34618329753066434430082904643, 1.30547658313554125066135762741, 2.84627170779629514049792507910, 3.98790655479432617874243898483, 4.99564851081204143336074448283, 6.39578608794965735001766047804, 6.96452347471109430357799336007, 8.001714286136272953030197192842, 9.080308085647208133333061696921, 9.992841160989945380999400750933, 10.56763297379459000829969443204

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