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Properties

Label 2-504-56.37-c1-0-11
Degree $2$
Conductor $504$
Sign $-0.308 - 0.951i$
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  + (1.00 + 0.990i)2-s + (0.0372 + 1.99i)4-s + (−0.586 − 0.338i)5-s + (2.23 + 1.41i)7-s + (−1.94 + 2.05i)8-s + (−0.256 − 0.922i)10-s + (−1.44 + 0.835i)11-s + 1.28i·13-s + (0.857 + 3.64i)14-s + (−3.99 + 0.148i)16-s + (3.18 + 5.51i)17-s + (2.20 + 1.27i)19-s + (0.655 − 1.18i)20-s + (−2.28 − 0.590i)22-s + (−0.127 + 0.221i)23-s + ⋯
L(s)  = 1  + (0.713 + 0.700i)2-s + (0.0186 + 0.999i)4-s + (−0.262 − 0.151i)5-s + (0.845 + 0.534i)7-s + (−0.687 + 0.726i)8-s + (−0.0811 − 0.291i)10-s + (−0.436 + 0.251i)11-s + 0.355i·13-s + (0.229 + 0.973i)14-s + (−0.999 + 0.0372i)16-s + (0.771 + 1.33i)17-s + (0.506 + 0.292i)19-s + (0.146 − 0.265i)20-s + (−0.487 − 0.125i)22-s + (−0.0266 + 0.0461i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.308 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.308 - 0.951i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.308 - 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18243 + 1.62600i\)
\(L(\frac12)\) \(\approx\) \(1.18243 + 1.62600i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.00 - 0.990i)T \)
3 \( 1 \)
7 \( 1 + (-2.23 - 1.41i)T \)
good5 \( 1 + (0.586 + 0.338i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.44 - 0.835i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.28iT - 13T^{2} \)
17 \( 1 + (-3.18 - 5.51i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.20 - 1.27i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.127 - 0.221i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.27iT - 29T^{2} \)
31 \( 1 + (-2.14 - 3.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.62 + 3.24i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.43T + 41T^{2} \)
43 \( 1 - 5.48iT - 43T^{2} \)
47 \( 1 + (-4.73 + 8.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.91 - 2.83i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.74 + 5.04i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (13.2 + 7.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-13.6 + 7.88i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.48T + 71T^{2} \)
73 \( 1 + (1.43 + 2.48i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.09 + 10.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.63iT - 83T^{2} \)
89 \( 1 + (-3.40 + 5.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.477T + 97T^{2} \)
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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

−11.48135928233785392783283055957, −10.39936879951532850695739108578, −9.145082919273298867477834115440, −8.062802230858992813637253353806, −7.80862603457931092335634845224, −6.41366348608167385638145320937, −5.55438584071391571393475252787, −4.64584532314035201370957059499, −3.63231502241912161018852683386, −2.12353793411361916099450122575, 1.04667332721590517964197927988, 2.68957804639082598595473279712, 3.73730355541507260441419869108, 4.93449195616920302517578634836, 5.55885060668257784760278819741, 7.03789221392398293309352089688, 7.79084304442091319118011012340, 9.088567578184104471194224596740, 10.04931887236118040312438150377, 10.87502782736567599840311630264

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