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Properties

Label 2-504-63.47-c1-0-6
Degree $2$
Conductor $504$
Sign $0.0491 - 0.998i$
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  + (−1.14 + 1.29i)3-s − 0.0525·5-s + (2.44 + 1.01i)7-s + (−0.371 − 2.97i)9-s + 2.48i·11-s + (2.51 − 1.45i)13-s + (0.0602 − 0.0682i)15-s + (2.88 + 4.99i)17-s + (−2.92 − 1.69i)19-s + (−4.12 + 2.00i)21-s + 8.63i·23-s − 4.99·25-s + (4.29 + 2.93i)27-s + (−6.23 − 3.60i)29-s + (8.59 + 4.96i)31-s + ⋯
L(s)  = 1  + (−0.661 + 0.749i)3-s − 0.0235·5-s + (0.922 + 0.385i)7-s + (−0.123 − 0.992i)9-s + 0.749i·11-s + (0.698 − 0.403i)13-s + (0.0155 − 0.0176i)15-s + (0.700 + 1.21i)17-s + (−0.671 − 0.387i)19-s + (−0.899 + 0.436i)21-s + 1.79i·23-s − 0.999·25-s + (0.825 + 0.564i)27-s + (−1.15 − 0.668i)29-s + (1.54 + 0.890i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0491 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0491 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.0491 - 0.998i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.0491 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.861616 + 0.820275i\)
\(L(\frac12)\) \(\approx\) \(0.861616 + 0.820275i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.14 - 1.29i)T \)
7 \( 1 + (-2.44 - 1.01i)T \)
good5 \( 1 + 0.0525T + 5T^{2} \)
11 \( 1 - 2.48iT - 11T^{2} \)
13 \( 1 + (-2.51 + 1.45i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.88 - 4.99i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.92 + 1.69i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.63iT - 23T^{2} \)
29 \( 1 + (6.23 + 3.60i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.59 - 4.96i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.770 - 1.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.392 - 0.679i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.03 - 3.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.657 - 1.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.710 + 0.410i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.32 - 4.01i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.87 + 2.81i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.95 + 12.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.51iT - 71T^{2} \)
73 \( 1 + (-6.75 + 3.90i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.50 + 12.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.14 - 5.44i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.93 + 13.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.57 - 0.909i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.17183950861126860799340172352, −10.28892288305818636974550440479, −9.545780387089143833445713293367, −8.473932259497322291395191131927, −7.64294401043983111941622468681, −6.22969616398317490347720312662, −5.50626617256798669415719264686, −4.51417317443339295571237878385, −3.52352781378912110358451188257, −1.63660859594919116649232006569, 0.843776179150408132947693049704, 2.28382307446017710049269964461, 4.03601229899381284051570885071, 5.16444688251125515094215862570, 6.09030635557125158212434293135, 7.01933353499809162709578615505, 7.995830537434545487270359601573, 8.610421296538914361615391591357, 10.05101220529890150914797929688, 10.97151908292316482153186060422

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