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Properties

Label 2-504-21.17-c3-0-3
Degree $2$
Conductor $504$
Sign $-0.980 - 0.198i$
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  + (3.30 + 5.73i)5-s + (−4.31 + 18.0i)7-s + (−1.54 − 0.893i)11-s + 4.72i·13-s + (−23.8 + 41.3i)17-s + (−40.1 + 23.1i)19-s + (−30.2 + 17.4i)23-s + (40.5 − 70.3i)25-s + 48.3i·29-s + (−107. − 62.2i)31-s + (−117. + 34.8i)35-s + (−137. − 238. i)37-s − 37.3·41-s − 215.·43-s + (53.1 + 91.9i)47-s + ⋯
L(s)  = 1  + (0.295 + 0.512i)5-s + (−0.233 + 0.972i)7-s + (−0.0424 − 0.0244i)11-s + 0.100i·13-s + (−0.340 + 0.589i)17-s + (−0.484 + 0.279i)19-s + (−0.273 + 0.158i)23-s + (0.324 − 0.562i)25-s + 0.309i·29-s + (−0.624 − 0.360i)31-s + (−0.567 + 0.168i)35-s + (−0.612 − 1.06i)37-s − 0.142·41-s − 0.765·43-s + (0.164 + 0.285i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.198i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.980 - 0.198i$
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: \((2,\ 504,\ (\ :3/2),\ -0.980 - 0.198i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7961696189\)
\(L(\frac12)\) \(\approx\) \(0.7961696189\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.31 - 18.0i)T \)
good5 \( 1 + (-3.30 - 5.73i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (1.54 + 0.893i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 4.72iT - 2.19e3T^{2} \)
17 \( 1 + (23.8 - 41.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (40.1 - 23.1i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (30.2 - 17.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 48.3iT - 2.43e4T^{2} \)
31 \( 1 + (107. + 62.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (137. + 238. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 37.3T + 6.89e4T^{2} \)
43 \( 1 + 215.T + 7.95e4T^{2} \)
47 \( 1 + (-53.1 - 91.9i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-233. - 134. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (149. - 259. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (292. - 168. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (188. - 326. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 816. iT - 3.57e5T^{2} \)
73 \( 1 + (596. + 344. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (307. + 531. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.32e3T + 5.71e5T^{2} \)
89 \( 1 + (-480. - 832. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 449. iT - 9.12e5T^{2} \)
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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

−10.82196194063991921085211594872, −10.11278044583515554803822349323, −9.079124290116760678477004236651, −8.407943290884956776093670150259, −7.19234292374526515715794109081, −6.23102573257615645868468384644, −5.52299463472229512258356357520, −4.15395175561503251055826097924, −2.88022289829448117254299088949, −1.85734496551482698430309136263, 0.23377009637002771046020096735, 1.60078504220484281085775771969, 3.15915224137285267214877073788, 4.36247092944930123317222318111, 5.24845325167372546461903523970, 6.50411088073178657760310067328, 7.27963033509158634443848854940, 8.354647399175009978865564419674, 9.245835194361882705612596538386, 10.10192046216247493226664862430

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