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Properties

Label 2-504-56.3-c1-0-16
Degree $2$
Conductor $504$
Sign $0.997 + 0.0660i$
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  + (−1.13 + 0.849i)2-s + (0.557 − 1.92i)4-s + (−1.03 − 1.80i)5-s + (1.25 + 2.33i)7-s + (1.00 + 2.64i)8-s + (2.70 + 1.15i)10-s + (0.669 − 1.16i)11-s − 2.50·13-s + (−3.39 − 1.57i)14-s + (−3.37 − 2.14i)16-s + (2.78 + 1.60i)17-s + (3.55 − 2.05i)19-s + (−4.03 + 0.991i)20-s + (0.227 + 1.88i)22-s + (5.54 − 3.20i)23-s + ⋯
L(s)  = 1  + (−0.799 + 0.600i)2-s + (0.278 − 0.960i)4-s + (−0.464 − 0.805i)5-s + (0.473 + 0.880i)7-s + (0.353 + 0.935i)8-s + (0.855 + 0.364i)10-s + (0.201 − 0.349i)11-s − 0.694·13-s + (−0.907 − 0.420i)14-s + (−0.844 − 0.535i)16-s + (0.674 + 0.389i)17-s + (0.815 − 0.470i)19-s + (−0.902 + 0.221i)20-s + (0.0485 + 0.401i)22-s + (1.15 − 0.668i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.997 + 0.0660i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.997 + 0.0660i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.943305 - 0.0312043i\)
\(L(\frac12)\) \(\approx\) \(0.943305 - 0.0312043i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 - 0.849i)T \)
3 \( 1 \)
7 \( 1 + (-1.25 - 2.33i)T \)
good5 \( 1 + (1.03 + 1.80i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.669 + 1.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.50T + 13T^{2} \)
17 \( 1 + (-2.78 - 1.60i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.55 + 2.05i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.54 + 3.20i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.66iT - 29T^{2} \)
31 \( 1 + (-2.21 + 3.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.50 + 3.17i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.55iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-0.565 - 0.980i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.43 - 4.29i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.29 + 3.63i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.57 - 4.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.93 + 6.81i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.29iT - 71T^{2} \)
73 \( 1 + (-0.480 - 0.277i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.26 + 3.04i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.503iT - 83T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.2iT - 97T^{2} \)
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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

−10.87995680595514057759816476130, −9.667293649216559775874612094851, −9.049121450731217144348001598534, −8.205640616663383426612991789226, −7.60771443561149449334633530919, −6.33092253115123926994398138827, −5.35501156733992160551692282054, −4.56294233296106963997425477262, −2.58186351391932834692911536963, −0.917585376354607934926928147675, 1.23176859484639774419194811427, 2.90385913708044036383991387777, 3.78419804076341094503810531011, 5.08365972745931070070348791937, 7.03217455804050753125526478442, 7.23963580028442537957660652816, 8.168010673474880680559514432038, 9.379093075456384834056109837765, 10.13186203908449679805276880424, 10.87038835077681679682191559847

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