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Properties

Label 2-504-21.17-c3-0-22
Degree $2$
Conductor $504$
Sign $-0.998 - 0.0499i$
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  + (−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

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0499i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.998 - 0.0499i$
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: \((2,\ 504,\ (\ :3/2),\ -0.998 - 0.0499i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7339193204\)
\(L(\frac12)\) \(\approx\) \(0.7339193204\)
\(L(\frac{5}{2})\) 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 \( 1 - 893. iT - 9.12e5T^{2} \)
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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

−10.29566540307007925687954716044, −9.018049315433014408448458142563, −8.256966259074254325712292722374, −7.44128060558661607863591825772, −6.42273693089441043200024791670, −5.12243768691317411470043933160, −4.30156042422513625662230279639, −3.23446215565601303013513829811, −1.41445269370377873885897441507, −0.22978051100743609279816367033, 1.88165550335170373594385424248, 3.02081474055623534268483788458, 4.18588966736660344170874358201, 5.37730090003196177860065031478, 6.53601650154206756270443896498, 7.14118810871221284163893507231, 8.441567392362358909741184793373, 9.053938172978986607746506097305, 10.07620292212415076991650751657, 11.30271997774197777747266124048

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