Properties

Label 2-504-21.17-c3-0-13
Degree $2$
Conductor $504$
Sign $0.732 - 0.681i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (10.9 + 18.9i)5-s + (−12.1 − 13.9i)7-s + (45.1 + 26.0i)11-s − 54.9i·13-s + (40.8 − 70.7i)17-s + (113. − 65.6i)19-s + (38.0 − 21.9i)23-s + (−176. + 305. i)25-s + 238. i·29-s + (174. + 100. i)31-s + (132. − 382. i)35-s + (12.0 + 20.8i)37-s − 102.·41-s + 119.·43-s + (20.2 + 35.1i)47-s + ⋯
L(s)  = 1  + (0.977 + 1.69i)5-s + (−0.655 − 0.755i)7-s + (1.23 + 0.714i)11-s − 1.17i·13-s + (0.582 − 1.00i)17-s + (1.37 − 0.792i)19-s + (0.345 − 0.199i)23-s + (−1.40 + 2.44i)25-s + 1.52i·29-s + (1.00 + 0.582i)31-s + (0.638 − 1.84i)35-s + (0.0534 + 0.0925i)37-s − 0.389·41-s + 0.424·43-s + (0.0629 + 0.109i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.681i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.732 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.732 - 0.681i$
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: \((2,\ 504,\ (\ :3/2),\ 0.732 - 0.681i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.481108682\)
\(L(\frac12)\) \(\approx\) \(2.481108682\)
\(L(\frac{5}{2})\) 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 \( 1 + (12.1 + 13.9i)T \)
good5 \( 1 + (-10.9 - 18.9i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-45.1 - 26.0i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 54.9iT - 2.19e3T^{2} \)
17 \( 1 + (-40.8 + 70.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-113. + 65.6i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-38.0 + 21.9i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 238. iT - 2.43e4T^{2} \)
31 \( 1 + (-174. - 100. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-12.0 - 20.8i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 102.T + 6.89e4T^{2} \)
43 \( 1 - 119.T + 7.95e4T^{2} \)
47 \( 1 + (-20.2 - 35.1i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (297. + 172. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (142. - 246. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (386. - 223. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-113. + 196. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 886. iT - 3.57e5T^{2} \)
73 \( 1 + (-6.46 - 3.73i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (404. + 700. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 943.T + 5.71e5T^{2} \)
89 \( 1 + (-575. - 996. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.55e3iT - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.45358969241387075927557168030, −9.851534573691085301255418668961, −9.236454371300399600766813682580, −7.39617861455696987256319303738, −7.02876922863348840712190475299, −6.21996866374522581598894977607, −5.05925354691474937589084777628, −3.35887229622870502519699766555, −2.87735534851631908914452962201, −1.15934666237594962032431588626, 0.967171093570407781296498826268, 1.92339431732729491102640417069, 3.64541472073819888896870161400, 4.76147096571807043553279909994, 6.02688030662826578349693869900, 6.13062221356758089370864844913, 7.979304895337146450228994143505, 8.885293534400893959829616448670, 9.407456179240836804766528945881, 9.959903474789717179950388380935

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