↑

Properties

Label 2-504-21.17-c3-0-21
Degree $2$
Conductor $504$
Sign $-0.380 + 0.924i$
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  + (5.00 + 8.66i)5-s + (1.56 − 18.4i)7-s + (−8.94 − 5.16i)11-s − 52.4i·13-s + (−0.584 + 1.01i)17-s + (−86.7 + 50.0i)19-s + (−90.1 + 52.0i)23-s + (12.4 − 21.6i)25-s − 187. i·29-s + (−107. − 62.0i)31-s + (167. − 78.6i)35-s + (16.0 + 27.7i)37-s + 415.·41-s − 193.·43-s + (−196. − 340. i)47-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s + (0.0847 − 0.996i)7-s + (−0.245 − 0.141i)11-s − 1.11i·13-s + (−0.00833 + 0.0144i)17-s + (−1.04 + 0.604i)19-s + (−0.817 + 0.471i)23-s + (0.0999 − 0.173i)25-s − 1.20i·29-s + (−0.622 − 0.359i)31-s + (0.809 − 0.379i)35-s + (0.0712 + 0.123i)37-s + 1.58·41-s − 0.685·43-s + (−0.609 − 1.05i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.380 + 0.924i$
Analytic conductor: \(29.7369\)
Root analytic conductor: \(5.45316\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :3/2),\ -0.380 + 0.924i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.186788060\)
\(L(\frac12)\) \(\approx\) \(1.186788060\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.56 + 18.4i)T \)
good5 \( 1 + (-5.00 - 8.66i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (8.94 + 5.16i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 52.4iT - 2.19e3T^{2} \)
17 \( 1 + (0.584 - 1.01i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (86.7 - 50.0i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (90.1 - 52.0i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 187. iT - 2.43e4T^{2} \)
31 \( 1 + (107. + 62.0i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-16.0 - 27.7i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 415.T + 6.89e4T^{2} \)
43 \( 1 + 193.T + 7.95e4T^{2} \)
47 \( 1 + (196. + 340. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-74.7 - 43.1i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-102. + 177. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (183. - 105. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-364. + 631. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 315. iT - 3.57e5T^{2} \)
73 \( 1 + (899. + 519. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (607. + 1.05e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 333.T + 5.71e5T^{2} \)
89 \( 1 + (-168. - 291. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 893. iT - 9.12e5T^{2} \)
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

−10.41029127430161410848850190428, −9.615907541750154219685659008478, −8.190896736437808067741045087195, −7.59438770685790348355303214420, −6.47589221712673183314602508666, −5.72644149074349710528081870761, −4.33538466556304840620367022339, −3.29412411971141608371411532685, −2.02958341346909722080260527506, −0.34911447863628258879402864128, 1.55351290450215329868413847708, 2.57580623020460167992540821608, 4.24777963958227452568202860435, 5.14421826722710466060913254505, 6.06386942077822994170347948703, 7.05757098307999344172074069239, 8.422266437450592815323618692759, 8.957298494178413082193120220979, 9.672720550305189205861041526882, 10.85493236656243724588945311118

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