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Properties

Label 2-504-56.13-c2-0-51
Degree $2$
Conductor $504$
Sign $0.638 - 0.769i$
Analytic cond. $13.7330$
Root an. cond. $3.70580$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.172 + 1.99i)2-s + (−3.94 − 0.688i)4-s + 6.16·5-s + (6.35 + 2.93i)7-s + (2.05 − 7.73i)8-s + (−1.06 + 12.2i)10-s − 6.45i·11-s + 5.92·13-s + (−6.94 + 12.1i)14-s + (15.0 + 5.42i)16-s − 29.7i·17-s − 7.53·19-s + (−24.3 − 4.24i)20-s + (12.8 + 1.11i)22-s + 40.1·23-s + ⋯
L(s)  = 1  + (−0.0863 + 0.996i)2-s + (−0.985 − 0.172i)4-s + 1.23·5-s + (0.907 + 0.419i)7-s + (0.256 − 0.966i)8-s + (−0.106 + 1.22i)10-s − 0.587i·11-s + 0.455·13-s + (−0.496 + 0.868i)14-s + (0.940 + 0.339i)16-s − 1.74i·17-s − 0.396·19-s + (−1.21 − 0.212i)20-s + (0.585 + 0.0507i)22-s + 1.74·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.638 - 0.769i$
Analytic conductor: \(13.7330\)
Root analytic conductor: \(3.70580\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1),\ 0.638 - 0.769i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.147282136\)
\(L(\frac12)\) \(\approx\) \(2.147282136\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.172 - 1.99i)T \)
3 \( 1 \)
7 \( 1 + (-6.35 - 2.93i)T \)
good5 \( 1 - 6.16T + 25T^{2} \)
11 \( 1 + 6.45iT - 121T^{2} \)
13 \( 1 - 5.92T + 169T^{2} \)
17 \( 1 + 29.7iT - 289T^{2} \)
19 \( 1 + 7.53T + 361T^{2} \)
23 \( 1 - 40.1T + 529T^{2} \)
29 \( 1 + 19.5iT - 841T^{2} \)
31 \( 1 + 20.7iT - 961T^{2} \)
37 \( 1 - 43.3iT - 1.36e3T^{2} \)
41 \( 1 + 43.0iT - 1.68e3T^{2} \)
43 \( 1 - 27.7iT - 1.84e3T^{2} \)
47 \( 1 - 75.5iT - 2.20e3T^{2} \)
53 \( 1 - 36.6iT - 2.80e3T^{2} \)
59 \( 1 - 21.9T + 3.48e3T^{2} \)
61 \( 1 + 49.7T + 3.72e3T^{2} \)
67 \( 1 - 106. iT - 4.48e3T^{2} \)
71 \( 1 - 84.5T + 5.04e3T^{2} \)
73 \( 1 + 67.2iT - 5.32e3T^{2} \)
79 \( 1 - 19.2T + 6.24e3T^{2} \)
83 \( 1 + 122.T + 6.88e3T^{2} \)
89 \( 1 + 43.7iT - 7.92e3T^{2} \)
97 \( 1 - 151. iT - 9.40e3T^{2} \)
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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

−10.72745470654256164149444771885, −9.551252962499369135981693304837, −9.069008027812694540503044727809, −8.157757534086102909018125837701, −7.12297104968357242858184506594, −6.11088989739380895924320652755, −5.38838492362856542496654445646, −4.57515809362258491520762360051, −2.76176181094142539692989534751, −1.11241720124494423057921985208, 1.35017212367818165372283573978, 2.09605337065911508902666683870, 3.63201643948885436387926837979, 4.78180556714310686412034483027, 5.63624885200888138128969625312, 6.92726606848720366021865415802, 8.268737074833550741576865994424, 8.927847448517102968359811138626, 9.926703327279541066708990522455, 10.66774156564445083402515090075

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