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Properties

Label 2-504-21.5-c3-0-12
Degree $2$
Conductor $504$
Sign $-0.124 + 0.992i$
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  + (−9.74 + 16.8i)5-s + (−18.4 − 1.67i)7-s + (−15.1 + 8.72i)11-s + 16.5i·13-s + (68.6 + 118. i)17-s + (−20.6 − 11.9i)19-s + (−108. − 62.8i)23-s + (−127. − 220. i)25-s − 105. i·29-s + (−95.0 + 54.8i)31-s + (207. − 294. i)35-s + (58.5 − 101. i)37-s + 348.·41-s − 141.·43-s + (172. − 299. i)47-s + ⋯
L(s)  = 1  + (−0.871 + 1.50i)5-s + (−0.995 − 0.0902i)7-s + (−0.414 + 0.239i)11-s + 0.352i·13-s + (0.979 + 1.69i)17-s + (−0.249 − 0.144i)19-s + (−0.987 − 0.570i)23-s + (−1.01 − 1.76i)25-s − 0.673i·29-s + (−0.550 + 0.318i)31-s + (1.00 − 1.42i)35-s + (0.260 − 0.450i)37-s + 1.32·41-s − 0.503·43-s + (0.536 − 0.929i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.124 + 0.992i$
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: \((2,\ 504,\ (\ :3/2),\ -0.124 + 0.992i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1052535662\)
\(L(\frac12)\) \(\approx\) \(0.1052535662\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.4 + 1.67i)T \)
good5 \( 1 + (9.74 - 16.8i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (15.1 - 8.72i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 16.5iT - 2.19e3T^{2} \)
17 \( 1 + (-68.6 - 118. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (20.6 + 11.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (108. + 62.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 105. iT - 2.43e4T^{2} \)
31 \( 1 + (95.0 - 54.8i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-58.5 + 101. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 348.T + 6.89e4T^{2} \)
43 \( 1 + 141.T + 7.95e4T^{2} \)
47 \( 1 + (-172. + 299. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-149. + 86.1i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (297. + 515. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (561. + 324. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (64.9 + 112. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 908. iT - 3.57e5T^{2} \)
73 \( 1 + (44.8 - 25.9i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-474. + 822. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.30e3T + 5.71e5T^{2} \)
89 \( 1 + (210. - 364. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 553. iT - 9.12e5T^{2} \)
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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

−10.44586160214402138389193331497, −9.625214005389821017963367459562, −8.230535582229792706933217654162, −7.51411526812717490305875858953, −6.58563939091580267630267298754, −5.92670750767398190646781665775, −4.08479681774101290510118190951, −3.44634814756469440176387929394, −2.29681401642471427892755360944, −0.03974704171202348532951546776, 0.984738921736881758344738113520, 2.93946659423737329553369789290, 4.01738752081339880088602892242, 5.10131213435572787511298735228, 5.89903356430026822608571626403, 7.41478198686199793206210039278, 7.972725560621998801225338585701, 9.116239403727587123262004274711, 9.564523389002662931452422763502, 10.75894610876715449195774195720

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