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Properties

Label 2-504-56.27-c1-0-19
Degree $2$
Conductor $504$
Sign $0.493 - 0.869i$
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  + (0.185 + 1.40i)2-s + (−1.93 + 0.520i)4-s + 3.84·5-s + (−1.62 − 2.09i)7-s + (−1.08 − 2.61i)8-s + (0.713 + 5.38i)10-s + 4.54·11-s + 1.81·13-s + (2.63 − 2.66i)14-s + (3.45 − 2.00i)16-s + 3.49i·17-s + 1.68i·19-s + (−7.42 + 2.00i)20-s + (0.843 + 6.37i)22-s − 5.00i·23-s + ⋯
L(s)  = 1  + (0.131 + 0.991i)2-s + (−0.965 + 0.260i)4-s + 1.71·5-s + (−0.612 − 0.790i)7-s + (−0.384 − 0.923i)8-s + (0.225 + 1.70i)10-s + 1.37·11-s + 0.503·13-s + (0.702 − 0.711i)14-s + (0.864 − 0.502i)16-s + 0.846i·17-s + 0.387i·19-s + (−1.66 + 0.447i)20-s + (0.179 + 1.35i)22-s − 1.04i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.493 - 0.869i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.493 - 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57906 + 0.919222i\)
\(L(\frac12)\) \(\approx\) \(1.57906 + 0.919222i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.185 - 1.40i)T \)
3 \( 1 \)
7 \( 1 + (1.62 + 2.09i)T \)
good5 \( 1 - 3.84T + 5T^{2} \)
11 \( 1 - 4.54T + 11T^{2} \)
13 \( 1 - 1.81T + 13T^{2} \)
17 \( 1 - 3.49iT - 17T^{2} \)
19 \( 1 - 1.68iT - 19T^{2} \)
23 \( 1 + 5.00iT - 23T^{2} \)
29 \( 1 - 1.81iT - 29T^{2} \)
31 \( 1 + 5.34T + 31T^{2} \)
37 \( 1 + 1.42iT - 37T^{2} \)
41 \( 1 - 8.97iT - 41T^{2} \)
43 \( 1 + 8.03T + 43T^{2} \)
47 \( 1 - 4.83T + 47T^{2} \)
53 \( 1 + 5.87iT - 53T^{2} \)
59 \( 1 - 8.46iT - 59T^{2} \)
61 \( 1 + 3.01T + 61T^{2} \)
67 \( 1 + 4.42T + 67T^{2} \)
71 \( 1 + 1.47iT - 71T^{2} \)
73 \( 1 + 6.98iT - 73T^{2} \)
79 \( 1 + 2.97iT - 79T^{2} \)
83 \( 1 + 10.5iT - 83T^{2} \)
89 \( 1 + 15.9iT - 89T^{2} \)
97 \( 1 - 11.6iT - 97T^{2} \)
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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

−10.69847595843084500711807063060, −9.951645729640171210103729747950, −9.248916188606558066246900119334, −8.505211873159884959921228556881, −7.07643930315871644158326321450, −6.32798741958628724649531416106, −5.89713560648044873780349165775, −4.52624231428096023554513590254, −3.43053417073925338316653815348, −1.43953232218256369428386383853, 1.46885040303660548096596310904, 2.51342495763389521010352807875, 3.67422346387292718808450997182, 5.21265791084918981739396099189, 5.87592921318864147446550549886, 6.80121045561062831943674303330, 8.700908486661352640699022341893, 9.410588666274874098401828006847, 9.615203492683142888686496551724, 10.75633780035789811311083667156

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