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Properties

Label 2-504-56.3-c1-0-9
Degree $2$
Conductor $504$
Sign $0.998 + 0.0580i$
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  + (0.297 − 1.38i)2-s + (−1.82 − 0.821i)4-s + (1.44 + 2.49i)5-s + (2.63 + 0.194i)7-s + (−1.67 + 2.27i)8-s + (3.88 − 1.25i)10-s + (−2.91 + 5.04i)11-s − 1.04·13-s + (1.05 − 3.59i)14-s + (2.64 + 2.99i)16-s + (5.91 + 3.41i)17-s + (−0.589 + 0.340i)19-s + (−0.577 − 5.73i)20-s + (6.11 + 5.52i)22-s + (−1.85 + 1.07i)23-s + ⋯
L(s)  = 1  + (0.210 − 0.977i)2-s + (−0.911 − 0.410i)4-s + (0.644 + 1.11i)5-s + (0.997 + 0.0733i)7-s + (−0.593 + 0.805i)8-s + (1.22 − 0.395i)10-s + (−0.878 + 1.52i)11-s − 0.290·13-s + (0.281 − 0.959i)14-s + (0.662 + 0.749i)16-s + (1.43 + 0.827i)17-s + (−0.135 + 0.0781i)19-s + (−0.129 − 1.28i)20-s + (1.30 + 1.17i)22-s + (−0.387 + 0.223i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.998 + 0.0580i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.998 + 0.0580i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63495 - 0.0475119i\)
\(L(\frac12)\) \(\approx\) \(1.63495 - 0.0475119i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.297 + 1.38i)T \)
3 \( 1 \)
7 \( 1 + (-2.63 - 0.194i)T \)
good5 \( 1 + (-1.44 - 2.49i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.91 - 5.04i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.04T + 13T^{2} \)
17 \( 1 + (-5.91 - 3.41i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.589 - 0.340i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.85 - 1.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.61iT - 29T^{2} \)
31 \( 1 + (-1.91 + 3.31i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.06 + 1.19i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.19iT - 41T^{2} \)
43 \( 1 - 1.34T + 43T^{2} \)
47 \( 1 + (5.52 + 9.57i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.99 - 4.03i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.81 + 3.93i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.63 + 2.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.65 - 11.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.08iT - 71T^{2} \)
73 \( 1 + (-4.88 - 2.82i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.9 + 6.32i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.482iT - 83T^{2} \)
89 \( 1 + (10.7 - 6.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.63iT - 97T^{2} \)
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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

−10.75473966578212399370450299614, −10.13722942763217854295755598852, −9.698503323924311419695818785338, −8.185735458216402904128920520811, −7.45516749647698729795640229679, −6.01396774523748128093812581280, −5.12914967661088477889243053969, −4.03633715058045828833638283322, −2.59728330700092459297930900115, −1.84686403812821232929346670783, 1.01592912306914967560307849165, 3.14377395655631368906010153671, 4.75317063850061658011851252830, 5.28018307720909758390443206260, 6.01409627452214148538228111360, 7.48798660189836007105040382533, 8.220055254181327654411342228291, 8.836701739935695108544629285767, 9.781213236716217903619419284389, 10.87086518643652469356006547680

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