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Properties

Label 2-504-56.19-c1-0-17
Degree $2$
Conductor $504$
Sign $-0.0686 - 0.997i$
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  + (1.04 + 0.949i)2-s + (0.195 + 1.99i)4-s + (−0.155 + 0.268i)5-s + (2.58 + 0.560i)7-s + (−1.68 + 2.27i)8-s + (−0.418 + 0.134i)10-s + (0.622 + 1.07i)11-s + 2.68·13-s + (2.17 + 3.04i)14-s + (−3.92 + 0.778i)16-s + (−1.93 + 1.11i)17-s + (−5.14 − 2.96i)19-s + (−0.565 − 0.256i)20-s + (−0.371 + 1.72i)22-s + (2.86 + 1.65i)23-s + ⋯
L(s)  = 1  + (0.740 + 0.671i)2-s + (0.0978 + 0.995i)4-s + (−0.0694 + 0.120i)5-s + (0.977 + 0.211i)7-s + (−0.595 + 0.803i)8-s + (−0.132 + 0.0424i)10-s + (0.187 + 0.325i)11-s + 0.745·13-s + (0.581 + 0.813i)14-s + (−0.980 + 0.194i)16-s + (−0.468 + 0.270i)17-s + (−1.17 − 0.681i)19-s + (−0.126 − 0.0573i)20-s + (−0.0792 + 0.366i)22-s + (0.596 + 0.344i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0686 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0686 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.0686 - 0.997i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.0686 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51941 + 1.62764i\)
\(L(\frac12)\) \(\approx\) \(1.51941 + 1.62764i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.04 - 0.949i)T \)
3 \( 1 \)
7 \( 1 + (-2.58 - 0.560i)T \)
good5 \( 1 + (0.155 - 0.268i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.622 - 1.07i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.68T + 13T^{2} \)
17 \( 1 + (1.93 - 1.11i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.14 + 2.96i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.86 - 1.65i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.191iT - 29T^{2} \)
31 \( 1 + (-1.95 - 3.38i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.643 - 0.371i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.28iT - 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + (-5.43 + 9.42i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.8 + 6.24i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.16 - 2.98i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.58 + 7.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.25 + 3.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.92iT - 71T^{2} \)
73 \( 1 + (-6.97 + 4.02i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (13.2 + 7.66i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.2iT - 83T^{2} \)
89 \( 1 + (-5.53 - 3.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.62iT - 97T^{2} \)
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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

−11.30825108568173047443805957670, −10.54803734558842228705914783322, −8.836191968956356762532713495900, −8.542518232187548030831990944296, −7.30308035886666206034793825299, −6.60480659558021987782027488275, −5.43780100514408673216295590181, −4.61971069812558472366232315644, −3.56530996843669710598831843135, −2.06653104984937091987762941566, 1.21368994298054381176256824244, 2.58364068388201677837446209229, 4.02591129790854981759052396291, 4.69622953258827755198470057716, 5.88376185498958462253932021672, 6.74515679800611648382529713701, 8.170783036131524697665863934720, 8.902203469720276356400892026374, 10.15206590782653236550190732513, 10.90830767618973971686015636825

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