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Properties

Label 2-504-8.3-c2-0-7
Degree $2$
Conductor $504$
Sign $-0.467 + 0.883i$
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  + (0.707 + 1.87i)2-s + (−3 + 2.64i)4-s + 9.03i·5-s − 2.64i·7-s + (−7.07 − 3.74i)8-s + (−16.8 + 6.38i)10-s − 12.4·11-s + 9.03i·13-s + (4.94 − 1.87i)14-s + (1.99 − 15.8i)16-s − 12.3·17-s + 28.8·19-s + (−23.8 − 27.0i)20-s + (−8.82 − 23.3i)22-s − 24.6i·23-s + ⋯
L(s)  = 1  + (0.353 + 0.935i)2-s + (−0.750 + 0.661i)4-s + 1.80i·5-s − 0.377i·7-s + (−0.883 − 0.467i)8-s + (−1.68 + 0.638i)10-s − 1.13·11-s + 0.694i·13-s + (0.353 − 0.133i)14-s + (0.124 − 0.992i)16-s − 0.726·17-s + 1.51·19-s + (−1.19 − 1.35i)20-s + (−0.401 − 1.06i)22-s − 1.07i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 + 0.883i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.467 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.467 + 0.883i$
Analytic conductor: \(13.7330\)
Root analytic conductor: \(3.70580\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1),\ -0.467 + 0.883i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9207702480\)
\(L(\frac12)\) \(\approx\) \(0.9207702480\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.87i)T \)
3 \( 1 \)
7 \( 1 + 2.64iT \)
good5 \( 1 - 9.03iT - 25T^{2} \)
11 \( 1 + 12.4T + 121T^{2} \)
13 \( 1 - 9.03iT - 169T^{2} \)
17 \( 1 + 12.3T + 289T^{2} \)
19 \( 1 - 28.8T + 361T^{2} \)
23 \( 1 + 24.6iT - 529T^{2} \)
29 \( 1 - 22.4iT - 841T^{2} \)
31 \( 1 - 16.7iT - 961T^{2} \)
37 \( 1 + 16.2iT - 1.36e3T^{2} \)
41 \( 1 + 6.97T + 1.68e3T^{2} \)
43 \( 1 + 22.8T + 1.84e3T^{2} \)
47 \( 1 + 6.19iT - 2.20e3T^{2} \)
53 \( 1 - 8.01iT - 2.80e3T^{2} \)
59 \( 1 + 30.4T + 3.48e3T^{2} \)
61 \( 1 - 15.2iT - 3.72e3T^{2} \)
67 \( 1 + 78.6T + 4.48e3T^{2} \)
71 \( 1 + 17.5iT - 5.04e3T^{2} \)
73 \( 1 - 46.6T + 5.32e3T^{2} \)
79 \( 1 - 81.0iT - 6.24e3T^{2} \)
83 \( 1 + 40.3T + 6.88e3T^{2} \)
89 \( 1 + 111.T + 7.92e3T^{2} \)
97 \( 1 + 164.T + 9.40e3T^{2} \)
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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

−11.17555873569065271579861148218, −10.45083920475618501417422619153, −9.554398931564887762052329362139, −8.324384894548178986041347723046, −7.27653947562470879210188108019, −6.94707120769901225549838414961, −5.95312702469145557087826519446, −4.79711789505409460074473462068, −3.52649028332714898955434885055, −2.63590115193779445364392690569, 0.32296510087701871169410541065, 1.61207137594965906829614906486, 3.01701105617034633940617561011, 4.38899446260706918651556428068, 5.26118311518808628633465218659, 5.71555586124520424473435209145, 7.77112536912234569533824366697, 8.503025621720802950912268259412, 9.425260351323366258170791148138, 9.968937950178690995910510683831

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