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Properties

Label 2-504-63.4-c1-0-14
Degree $2$
Conductor $504$
Sign $-0.0117 + 0.999i$
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  + (0.134 + 1.72i)3-s − 3.43·5-s + (−1.83 − 1.90i)7-s + (−2.96 + 0.465i)9-s + 4.40·11-s + (1.49 − 2.58i)13-s + (−0.463 − 5.93i)15-s + (0.542 − 0.939i)17-s + (−3.74 − 6.48i)19-s + (3.03 − 3.43i)21-s − 4.32·23-s + 6.80·25-s + (−1.20 − 5.05i)27-s + (1.68 + 2.91i)29-s + (−4.68 − 8.11i)31-s + ⋯
L(s)  = 1  + (0.0778 + 0.996i)3-s − 1.53·5-s + (−0.695 − 0.718i)7-s + (−0.987 + 0.155i)9-s + 1.32·11-s + (0.414 − 0.717i)13-s + (−0.119 − 1.53i)15-s + (0.131 − 0.227i)17-s + (−0.858 − 1.48i)19-s + (0.662 − 0.748i)21-s − 0.901·23-s + 1.36·25-s + (−0.231 − 0.972i)27-s + (0.312 + 0.541i)29-s + (−0.841 − 1.45i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0117 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0117 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.0117 + 0.999i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.0117 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.355659 - 0.359848i\)
\(L(\frac12)\) \(\approx\) \(0.355659 - 0.359848i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.134 - 1.72i)T \)
7 \( 1 + (1.83 + 1.90i)T \)
good5 \( 1 + 3.43T + 5T^{2} \)
11 \( 1 - 4.40T + 11T^{2} \)
13 \( 1 + (-1.49 + 2.58i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.542 + 0.939i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.74 + 6.48i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.32T + 23T^{2} \)
29 \( 1 + (-1.68 - 2.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.68 + 8.11i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.50 + 4.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.20 - 2.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.31 - 5.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.50 - 2.60i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.530 - 0.919i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.20 + 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.71 + 4.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.66 + 2.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + (8.21 - 14.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.17 + 2.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.60 + 2.78i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.67 - 9.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.40 + 11.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.94317493758442887208106536414, −9.767197228044872293387165471204, −8.985220278611393354287418179718, −8.090284263776122031057360575020, −7.12020573432095837287446437247, −6.08475421542346128854362350467, −4.51012087756549218935830084681, −3.95010091092133130994014213100, −3.13586970706140982341412546218, −0.30260261316675032386054723476, 1.68356934504574092492070409025, 3.37642512557150950265793702942, 4.11050115885012472684710309063, 5.91480662713437460659163959268, 6.61647900684218167305744832640, 7.49472190564926706545133203174, 8.544659434991534651905958386993, 8.892045465965636292715592670249, 10.37678883925594575430170366024, 11.58643802055743220880071418021

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