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Properties

Label 2-504-63.47-c1-0-3
Degree $2$
Conductor $504$
Sign $-0.680 - 0.732i$
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  + (1.51 + 0.844i)3-s − 2.84·5-s + (−2.64 − 0.0704i)7-s + (1.57 + 2.55i)9-s + 2.45i·11-s + (−3.06 + 1.76i)13-s + (−4.30 − 2.40i)15-s + (2.91 + 5.05i)17-s + (−2.90 − 1.67i)19-s + (−3.94 − 2.33i)21-s + 8.01i·23-s + 3.10·25-s + (0.225 + 5.19i)27-s + (−1.45 − 0.839i)29-s + (−3.45 − 1.99i)31-s + ⋯
L(s)  = 1  + (0.873 + 0.487i)3-s − 1.27·5-s + (−0.999 − 0.0266i)7-s + (0.524 + 0.851i)9-s + 0.738i·11-s + (−0.848 + 0.490i)13-s + (−1.11 − 0.620i)15-s + (0.708 + 1.22i)17-s + (−0.665 − 0.384i)19-s + (−0.859 − 0.510i)21-s + 1.67i·23-s + 0.621·25-s + (0.0434 + 0.999i)27-s + (−0.269 − 0.155i)29-s + (−0.620 − 0.357i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 - 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.680 - 0.732i$
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: \((2,\ 504,\ (\ :1/2),\ -0.680 - 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.365432 + 0.837873i\)
\(L(\frac12)\) \(\approx\) \(0.365432 + 0.837873i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.51 - 0.844i)T \)
7 \( 1 + (2.64 + 0.0704i)T \)
good5 \( 1 + 2.84T + 5T^{2} \)
11 \( 1 - 2.45iT - 11T^{2} \)
13 \( 1 + (3.06 - 1.76i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.91 - 5.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.90 + 1.67i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.01iT - 23T^{2} \)
29 \( 1 + (1.45 + 0.839i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.45 + 1.99i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.07 + 7.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.43 + 9.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.27 + 5.67i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.31 - 5.74i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.64 + 4.41i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.178 - 0.309i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.52 + 1.45i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.14 - 12.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.96iT - 71T^{2} \)
73 \( 1 + (-5.42 + 3.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.75 - 9.96i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.189 + 0.327i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.05 - 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.00 + 2.31i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.16338238730662288977283092018, −10.17488573077469062334105827926, −9.479879770517706813380441116089, −8.631795269021309308245427048356, −7.52557413553494858870533308686, −7.15608289056387277577218351381, −5.51064316384306455742009958381, −4.03086413268102224782053171414, −3.74723598891045107783834870706, −2.27235225159437846153836604810, 0.47479944710862654210113515513, 2.76898796680718909114015914971, 3.42082574459826405814768373058, 4.60913011749373861902731029735, 6.21593861929559432104500673696, 7.14780183311070928187350301985, 7.893168922148350966491813333029, 8.633591095270509926870948175138, 9.599041819214448277549286565709, 10.48431988145345529079226128330

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