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Properties

Label 2-504-504.293-c1-0-18
Degree $2$
Conductor $504$
Sign $0.311 - 0.950i$
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  + (−1.22 + 0.707i)2-s + (1.68 + 0.420i)3-s + (0.999 − 1.73i)4-s + (−3.32 − 1.91i)5-s + (−2.35 + 0.672i)6-s + (1.32 + 2.29i)7-s + 2.82i·8-s + (2.64 + 1.41i)9-s + 5.42·10-s + (2.40 − 2.48i)12-s + (−2.27 + 3.94i)13-s + (−3.24 − 1.87i)14-s + (−4.77 − 4.61i)15-s + (−2.00 − 3.46i)16-s + (−4.24 + 0.138i)18-s + 7.01·19-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)2-s + (0.970 + 0.242i)3-s + (0.499 − 0.866i)4-s + (−1.48 − 0.857i)5-s + (−0.961 + 0.274i)6-s + (0.499 + 0.866i)7-s + 0.999i·8-s + (0.881 + 0.471i)9-s + 1.71·10-s + (0.695 − 0.718i)12-s + (−0.631 + 1.09i)13-s + (−0.866 − 0.499i)14-s + (−1.23 − 1.19i)15-s + (−0.500 − 0.866i)16-s + (−0.999 + 0.0327i)18-s + 1.60·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.311 - 0.950i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.311 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.874267 + 0.633730i\)
\(L(\frac12)\) \(\approx\) \(0.874267 + 0.633730i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
3 \( 1 + (-1.68 - 0.420i)T \)
7 \( 1 + (-1.32 - 2.29i)T \)
good5 \( 1 + (3.32 + 1.91i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.27 - 3.94i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 7.01T + 19T^{2} \)
23 \( 1 + (-7.74 - 4.46i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (4.58 + 2.64i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.66 + 6.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (8.67 + 15.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (15.7 - 9.08i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (48.5 - 84.0i)T^{2} \)
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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

−11.23854677412188127571030776881, −9.685926766084678448753560554044, −9.088026627687704499191556591947, −8.518553448039680971050639527414, −7.61882885204483420072179013089, −7.16131270010028167029386491790, −5.27508101141685110540405217360, −4.56521668956274354902811127598, −3.07384281375735320303187385676, −1.46509803086782960666644118065, 0.876640931442413628308369154133, 2.84982326597811260759921613293, 3.39731618480424375521925649563, 4.53544889419334890470451523966, 6.94382963390091701734665979495, 7.44091280166356580497315703377, 7.907835303302396024901513876421, 8.804326478008576400538913755553, 9.987691498179351766666603860516, 10.66431857554431226200632353668

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