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Properties

Label 2-504-7.2-c3-0-1
Degree $2$
Conductor $504$
Sign $-0.999 + 0.0165i$
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  + (0.0642 − 0.111i)5-s + (−0.866 + 18.4i)7-s + (27.0 + 46.7i)11-s − 50.2·13-s + (−65.7 − 113. i)17-s + (−45.7 + 79.2i)19-s + (89.7 − 155. i)23-s + (62.4 + 108. i)25-s + 69.8·29-s + (−163. − 283. i)31-s + (2.00 + 1.28i)35-s + (−150. + 261. i)37-s − 296.·41-s − 144.·43-s + (−180. + 311. i)47-s + ⋯
L(s)  = 1  + (0.00575 − 0.00996i)5-s + (−0.0467 + 0.998i)7-s + (0.740 + 1.28i)11-s − 1.07·13-s + (−0.937 − 1.62i)17-s + (−0.552 + 0.956i)19-s + (0.813 − 1.40i)23-s + (0.499 + 0.865i)25-s + 0.447·29-s + (−0.946 − 1.63i)31-s + (0.00968 + 0.00621i)35-s + (−0.670 + 1.16i)37-s − 1.12·41-s − 0.511·43-s + (−0.558 + 0.967i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0165i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.999 + 0.0165i$
Analytic conductor: \(29.7369\)
Root analytic conductor: \(5.45316\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :3/2),\ -0.999 + 0.0165i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4083711180\)
\(L(\frac12)\) \(\approx\) \(0.4083711180\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.866 - 18.4i)T \)
good5 \( 1 + (-0.0642 + 0.111i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-27.0 - 46.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 50.2T + 2.19e3T^{2} \)
17 \( 1 + (65.7 + 113. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (45.7 - 79.2i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-89.7 + 155. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 69.8T + 2.43e4T^{2} \)
31 \( 1 + (163. + 283. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (150. - 261. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 296.T + 6.89e4T^{2} \)
43 \( 1 + 144.T + 7.95e4T^{2} \)
47 \( 1 + (180. - 311. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-0.917 - 1.58i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (26.6 + 46.1i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-54.0 + 93.6i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (421. + 729. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 241.T + 3.57e5T^{2} \)
73 \( 1 + (-103. - 179. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (279. - 484. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 986.T + 5.71e5T^{2} \)
89 \( 1 + (221. - 383. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 740.T + 9.12e5T^{2} \)
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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

−11.05347884647043703283192532649, −9.777336687000041195355881365867, −9.351488777460715136764867366207, −8.379104596969021219405862289132, −7.15391130276681526439013053670, −6.54366660349666483806981779829, −5.13812422503943538746658352520, −4.49667351618393012049517775739, −2.84585612107551338962008005570, −1.88471995312322034625473680601, 0.12042461616607961757633782616, 1.56262320139951039023640068444, 3.21302154378945215369181046967, 4.15623297450929685877049873663, 5.28162197299250666413800505458, 6.58998244589849848146612088024, 7.10413787998908178415453405344, 8.447969676702076254940367037626, 8.999489201088580626086873207189, 10.28808640172095056417113959464

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