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Properties

Label 2-504-21.17-c3-0-11
Degree $2$
Conductor $504$
Sign $0.893 + 0.449i$
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  + (2.08 + 3.60i)5-s + (−18.2 − 2.97i)7-s + (−33.1 − 19.1i)11-s + 49.6i·13-s + (−7.65 + 13.2i)17-s + (122. − 70.9i)19-s + (136. − 78.9i)23-s + (53.8 − 93.2i)25-s + 204. i·29-s + (−90.5 − 52.2i)31-s + (−27.3 − 72.1i)35-s + (−194. − 336. i)37-s + 325.·41-s + 191.·43-s + (249. + 432. i)47-s + ⋯
L(s)  = 1  + (0.186 + 0.322i)5-s + (−0.986 − 0.160i)7-s + (−0.909 − 0.524i)11-s + 1.05i·13-s + (−0.109 + 0.189i)17-s + (1.48 − 0.856i)19-s + (1.24 − 0.715i)23-s + (0.430 − 0.745i)25-s + 1.31i·29-s + (−0.524 − 0.302i)31-s + (−0.131 − 0.348i)35-s + (−0.862 − 1.49i)37-s + 1.23·41-s + 0.678·43-s + (0.775 + 1.34i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.893 + 0.449i$
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: \((2,\ 504,\ (\ :3/2),\ 0.893 + 0.449i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.572637732\)
\(L(\frac12)\) \(\approx\) \(1.572637732\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.2 + 2.97i)T \)
good5 \( 1 + (-2.08 - 3.60i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (33.1 + 19.1i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 49.6iT - 2.19e3T^{2} \)
17 \( 1 + (7.65 - 13.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-122. + 70.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-136. + 78.9i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 204. iT - 2.43e4T^{2} \)
31 \( 1 + (90.5 + 52.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (194. + 336. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 325.T + 6.89e4T^{2} \)
43 \( 1 - 191.T + 7.95e4T^{2} \)
47 \( 1 + (-249. - 432. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-37.8 - 21.8i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (86.5 - 149. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-208. + 120. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-440. + 763. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 1.01e3iT - 3.57e5T^{2} \)
73 \( 1 + (361. + 208. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-237. - 411. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 652.T + 5.71e5T^{2} \)
89 \( 1 + (-298. - 517. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.77e3iT - 9.12e5T^{2} \)
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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

−10.67823735346469406317764814602, −9.357645836246545128212655452636, −8.985743264192567839290430112913, −7.49577596670123188769700159120, −6.84641334459289269606110668552, −5.85340003065313474703926213131, −4.76451882492380721440047264864, −3.39053973510934606921094812501, −2.51215720459050890152435111076, −0.64729815242982340844932643076, 0.925395900152119450687663899764, 2.65137797046200813064564386991, 3.56604112144580922860570485207, 5.18310878973310139326624772341, 5.67321407369986519476487252599, 7.05709124946083734548351311202, 7.75946458384250850733332952852, 8.918457767485978246833921430503, 9.793496477936077665015574114405, 10.31947441005772817558585568909

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