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Properties

Label 2-504-504.461-c1-0-66
Degree $2$
Conductor $504$
Sign $0.617 - 0.786i$
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  + (1.22 + 0.707i)2-s + (1.68 + 0.420i)3-s + (0.999 + 1.73i)4-s + (0.0658 − 0.0380i)5-s + (1.76 + 1.70i)6-s + (1.32 − 2.29i)7-s + 2.82i·8-s + (2.64 + 1.41i)9-s + 0.107·10-s + (0.951 + 3.33i)12-s + (−2.27 − 3.94i)13-s + (3.24 − 1.87i)14-s + (0.126 − 0.0361i)15-s + (−2.00 + 3.46i)16-s + (2.24 + 3.60i)18-s − 7.99·19-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (0.970 + 0.242i)3-s + (0.499 + 0.866i)4-s + (0.0294 − 0.0169i)5-s + (0.718 + 0.695i)6-s + (0.499 − 0.866i)7-s + 0.999i·8-s + (0.881 + 0.471i)9-s + 0.0339·10-s + (0.274 + 0.961i)12-s + (−0.631 − 1.09i)13-s + (0.866 − 0.499i)14-s + (0.0326 − 0.00933i)15-s + (−0.500 + 0.866i)16-s + (0.528 + 0.849i)18-s − 1.83·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.617 - 0.786i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.85226 + 1.38727i\)
\(L(\frac12)\) \(\approx\) \(2.85226 + 1.38727i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 + (-1.68 - 0.420i)T \)
7 \( 1 + (-1.32 + 2.29i)T \)
good5 \( 1 + (-0.0658 + 0.0380i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.27 + 3.94i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 7.99T + 19T^{2} \)
23 \( 1 + (-1.25 + 0.727i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-4.58 + 2.64i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.13 - 7.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 16.5iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (-6.02 + 10.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-15.7 - 9.08i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (48.5 + 84.0i)T^{2} \)
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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

−10.89985728089225797573540352658, −10.34838122633387132998992426210, −9.037731735108813550833983007761, −8.044673839653708606313874059841, −7.56132022100266184596201555642, −6.53731049678421869036158671701, −5.15080502327689093630831751840, −4.30886701778183859231789232284, −3.37553575837016223542816014192, −2.12841073081864353413478239169, 1.89549990871750761991194188534, 2.54189559958770775979578142921, 3.98867173455516231542405899835, 4.78612662520544374941666645541, 6.14817984567443228521646720057, 6.97225511723566638275119922606, 8.211621737338655844139951832071, 9.072597953808015972945051248197, 9.895144226790404235100004848214, 10.94126858144767261039474771919

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