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Properties

Label 2-504-56.13-c2-0-15
Degree $2$
Conductor $504$
Sign $0.995 - 0.0930i$
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  + (0.606 − 1.90i)2-s + (−3.26 − 2.31i)4-s − 5.53·5-s + (−5.95 − 3.68i)7-s + (−6.38 + 4.82i)8-s + (−3.35 + 10.5i)10-s + 16.1i·11-s + 14.1·13-s + (−10.6 + 9.11i)14-s + (5.32 + 15.0i)16-s − 7.47i·17-s + 28.5·19-s + (18.0 + 12.7i)20-s + (30.7 + 9.77i)22-s − 8.09·23-s + ⋯
L(s)  = 1  + (0.303 − 0.952i)2-s + (−0.816 − 0.577i)4-s − 1.10·5-s + (−0.850 − 0.526i)7-s + (−0.797 + 0.602i)8-s + (−0.335 + 1.05i)10-s + 1.46i·11-s + 1.08·13-s + (−0.759 + 0.651i)14-s + (0.332 + 0.943i)16-s − 0.439i·17-s + 1.50·19-s + (0.904 + 0.639i)20-s + (1.39 + 0.444i)22-s − 0.352·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0930i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.995 - 0.0930i$
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: \((2,\ 504,\ (\ :1),\ 0.995 - 0.0930i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9580645812\)
\(L(\frac12)\) \(\approx\) \(0.9580645812\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.606 + 1.90i)T \)
3 \( 1 \)
7 \( 1 + (5.95 + 3.68i)T \)
good5 \( 1 + 5.53T + 25T^{2} \)
11 \( 1 - 16.1iT - 121T^{2} \)
13 \( 1 - 14.1T + 169T^{2} \)
17 \( 1 + 7.47iT - 289T^{2} \)
19 \( 1 - 28.5T + 361T^{2} \)
23 \( 1 + 8.09T + 529T^{2} \)
29 \( 1 - 5.91iT - 841T^{2} \)
31 \( 1 + 27.0iT - 961T^{2} \)
37 \( 1 + 20.5iT - 1.36e3T^{2} \)
41 \( 1 - 49.1iT - 1.68e3T^{2} \)
43 \( 1 - 42.4iT - 1.84e3T^{2} \)
47 \( 1 - 48.6iT - 2.20e3T^{2} \)
53 \( 1 - 77.1iT - 2.80e3T^{2} \)
59 \( 1 + 40.7T + 3.48e3T^{2} \)
61 \( 1 + 10.9T + 3.72e3T^{2} \)
67 \( 1 - 116. iT - 4.48e3T^{2} \)
71 \( 1 - 104.T + 5.04e3T^{2} \)
73 \( 1 - 82.8iT - 5.32e3T^{2} \)
79 \( 1 + 44.7T + 6.24e3T^{2} \)
83 \( 1 - 65.8T + 6.88e3T^{2} \)
89 \( 1 + 134. iT - 7.92e3T^{2} \)
97 \( 1 + 37.4iT - 9.40e3T^{2} \)
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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

−10.90487708829074075111581352479, −9.815358544341809110095628304621, −9.375765428870410031982904957414, −8.003843986380005967906157264021, −7.19433427810060569743497739585, −5.94345938912580180892840164694, −4.56570428989541957137715179049, −3.87082299969407164527760686103, −2.89297882390392718675665711880, −1.11692496883000629252571140461, 0.42952231250060809349411230415, 3.41300201989666262988391302262, 3.62480108223807764184360842618, 5.27839360957465714845772776300, 6.07168704586211405750619661552, 6.95496849262749259037082307872, 8.092776359180900803450648000116, 8.569874898561716288044671959401, 9.485809800890780204739209401767, 10.79885891171007022250239064688

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