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Properties

Label 2-504-168.11-c1-0-6
Degree $2$
Conductor $504$
Sign $0.631 - 0.775i$
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  + (−1.33 + 0.467i)2-s + (1.56 − 1.24i)4-s + (−0.316 − 0.548i)5-s + (2.06 + 1.65i)7-s + (−1.50 + 2.39i)8-s + (0.679 + 0.584i)10-s + (−0.424 − 0.245i)11-s + 3.13i·13-s + (−3.52 − 1.24i)14-s + (0.882 − 3.90i)16-s + (0.987 + 0.569i)17-s + (−0.591 − 1.02i)19-s + (−1.18 − 0.462i)20-s + (0.681 + 0.128i)22-s + (2.80 + 4.85i)23-s + ⋯
L(s)  = 1  + (−0.943 + 0.330i)2-s + (0.781 − 0.624i)4-s + (−0.141 − 0.245i)5-s + (0.779 + 0.626i)7-s + (−0.530 + 0.847i)8-s + (0.214 + 0.184i)10-s + (−0.127 − 0.0738i)11-s + 0.868i·13-s + (−0.942 − 0.332i)14-s + (0.220 − 0.975i)16-s + (0.239 + 0.138i)17-s + (−0.135 − 0.234i)19-s + (−0.263 − 0.103i)20-s + (0.145 + 0.0274i)22-s + (0.584 + 1.01i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.631 - 0.775i$
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: \((2,\ 504,\ (\ :1/2),\ 0.631 - 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.884302 + 0.419997i\)
\(L(\frac12)\) \(\approx\) \(0.884302 + 0.419997i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 - 0.467i)T \)
3 \( 1 \)
7 \( 1 + (-2.06 - 1.65i)T \)
good5 \( 1 + (0.316 + 0.548i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.424 + 0.245i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.13iT - 13T^{2} \)
17 \( 1 + (-0.987 - 0.569i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.591 + 1.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.80 - 4.85i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.05T + 29T^{2} \)
31 \( 1 + (-5.40 - 3.11i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.53 - 3.77i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.06iT - 41T^{2} \)
43 \( 1 - 4.65T + 43T^{2} \)
47 \( 1 + (-4.80 - 8.32i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.10 + 1.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.12 - 5.26i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.1 - 5.87i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.11 + 5.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.48T + 71T^{2} \)
73 \( 1 + (5.35 - 9.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.49 + 1.43i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.1iT - 83T^{2} \)
89 \( 1 + (9.12 - 5.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.17T + 97T^{2} \)
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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

−10.95117578874404058504115214851, −10.07760786737589909944862024876, −8.982184484616983912351136489857, −8.550774153866599290418650488477, −7.57759266677554967231191223579, −6.63062164800906969963350813015, −5.56455589950070047280657925307, −4.58492046975730274343941080944, −2.70501988503058167946366190725, −1.36595900864521128869053427209, 0.926557510834253571813952347639, 2.51214710684701793257713227336, 3.72959371482492946053140668061, 5.07872346350901385872939919896, 6.51175197903019904547535975436, 7.43092966048613676330073288533, 8.120635289508892619622017043042, 8.942533553375614939814817464044, 10.18235088926114285896052238421, 10.60365518958896471090481694891

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