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Properties

Label 2-504-168.11-c1-0-8
Degree $2$
Conductor $504$
Sign $0.984 - 0.175i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.00 − 0.989i)2-s + (0.0398 + 1.99i)4-s + (0.635 + 1.09i)5-s + (2.64 − 0.106i)7-s + (1.93 − 2.05i)8-s + (0.447 − 1.73i)10-s + (−1.05 − 0.606i)11-s + 3.91i·13-s + (−2.77 − 2.50i)14-s + (−3.99 + 0.159i)16-s + (−5.05 − 2.91i)17-s + (3.80 + 6.58i)19-s + (−2.17 + 1.31i)20-s + (0.460 + 1.65i)22-s + (4.20 + 7.28i)23-s + ⋯
L(s)  = 1  + (−0.714 − 0.700i)2-s + (0.0199 + 0.999i)4-s + (0.284 + 0.491i)5-s + (0.999 − 0.0401i)7-s + (0.685 − 0.727i)8-s + (0.141 − 0.550i)10-s + (−0.316 − 0.182i)11-s + 1.08i·13-s + (−0.741 − 0.670i)14-s + (−0.999 + 0.0398i)16-s + (−1.22 − 0.707i)17-s + (0.872 + 1.51i)19-s + (−0.486 + 0.293i)20-s + (0.0981 + 0.352i)22-s + (0.877 + 1.51i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.984 - 0.175i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.984 - 0.175i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11551 + 0.0987467i\)
\(L(\frac12)\) \(\approx\) \(1.11551 + 0.0987467i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.00 + 0.989i)T \)
3 \( 1 \)
7 \( 1 + (-2.64 + 0.106i)T \)
good5 \( 1 + (-0.635 - 1.09i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.05 + 0.606i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.91iT - 13T^{2} \)
17 \( 1 + (5.05 + 2.91i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.80 - 6.58i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.20 - 7.28i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.09T + 29T^{2} \)
31 \( 1 + (2.14 + 1.23i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.24 + 3.02i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.01iT - 41T^{2} \)
43 \( 1 + 7.49T + 43T^{2} \)
47 \( 1 + (0.704 + 1.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.74 - 3.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.86 - 2.80i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.16 + 2.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.22 - 3.85i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.44T + 71T^{2} \)
73 \( 1 + (-6.19 + 10.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.204 + 0.117i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.7iT - 83T^{2} \)
89 \( 1 + (8.38 - 4.84i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.07T + 97T^{2} \)
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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

−11.09476855026708400636838088968, −10.05751362499323649538063943906, −9.296717417637092727680007510502, −8.373253067474546672030902638644, −7.50503440224005981985594540595, −6.63676187227410191544342600261, −5.12446540379316541003256906172, −3.97997256778183564329292959608, −2.64457110885411437804032057707, −1.50111455904599414376914930190, 0.964598571546547606385340030904, 2.50055443066189448126002474398, 4.77782060122426264241788299582, 5.12174456337050345479856529269, 6.44811286358858276126839726998, 7.34131537186370440430495590767, 8.452030178816535345500260516251, 8.745335501644613441135341661876, 9.916033750412260829588455749217, 10.81062538698371060120256472224

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