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Properties

Label 2-504-56.37-c1-0-6
Degree $2$
Conductor $504$
Sign $-0.157 - 0.987i$
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 + 0.465i)2-s + (1.56 − 1.24i)4-s + (0.937 + 0.541i)5-s + (−1.62 − 2.09i)7-s + (−1.51 + 2.38i)8-s + (−1.50 − 0.286i)10-s + (−4.52 + 2.61i)11-s + 4.43i·13-s + (3.13 + 2.03i)14-s + (0.910 − 3.89i)16-s + (3.65 + 6.33i)17-s + (2.42 + 1.39i)19-s + (2.14 − 0.317i)20-s + (4.82 − 5.59i)22-s + (1.51 − 2.62i)23-s + ⋯
L(s)  = 1  + (−0.944 + 0.329i)2-s + (0.783 − 0.621i)4-s + (0.419 + 0.242i)5-s + (−0.612 − 0.790i)7-s + (−0.535 + 0.844i)8-s + (−0.475 − 0.0906i)10-s + (−1.36 + 0.787i)11-s + 1.23i·13-s + (0.838 + 0.544i)14-s + (0.227 − 0.973i)16-s + (0.886 + 1.53i)17-s + (0.555 + 0.320i)19-s + (0.478 − 0.0708i)20-s + (1.02 − 1.19i)22-s + (0.315 − 0.546i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.157 - 0.987i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.157 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.468944 + 0.549598i\)
\(L(\frac12)\) \(\approx\) \(0.468944 + 0.549598i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 - 0.465i)T \)
3 \( 1 \)
7 \( 1 + (1.62 + 2.09i)T \)
good5 \( 1 + (-0.937 - 0.541i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.52 - 2.61i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.43iT - 13T^{2} \)
17 \( 1 + (-3.65 - 6.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.42 - 1.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.51 + 2.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.06iT - 29T^{2} \)
31 \( 1 + (-3.20 - 5.55i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.85 - 4.53i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.05T + 41T^{2} \)
43 \( 1 - 10.7iT - 43T^{2} \)
47 \( 1 + (-4.54 + 7.86i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.71 + 0.989i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.01 - 3.47i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.85 + 3.95i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.26 - 3.03i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.53T + 71T^{2} \)
73 \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.621 + 1.07i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.1iT - 83T^{2} \)
89 \( 1 + (3.02 - 5.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.82T + 97T^{2} \)
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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

−10.62876136099292248065367282268, −10.22216612006561623390288385171, −9.610962425035311068425949551948, −8.397858949767987376959376473781, −7.58483314005635391006980468463, −6.72036038314626099147277481110, −5.94524293697534573741312195907, −4.60812874478694659878419349331, −2.97134739031215570882176462964, −1.58160573580533612833718393668, 0.58348729836148805383237320781, 2.59855762953309073095810165074, 3.17876296725600715977370524098, 5.37355918976461399604297298178, 5.89166501609800566219398103600, 7.44292148597387311512208082184, 7.976945406690938336999803737159, 9.140642849584717637091347282114, 9.673993177450611955416702378757, 10.50093666672690409408900191425

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