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Properties

Label 2-504-63.16-c1-0-4
Degree $2$
Conductor $504$
Sign $-0.463 - 0.885i$
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  + (−1.61 + 0.618i)3-s − 1.83·5-s + (2.45 − 0.997i)7-s + (2.23 − 2.00i)9-s − 3.09·11-s + (2.40 + 4.16i)13-s + (2.97 − 1.13i)15-s + (1.87 + 3.24i)17-s + (−2.71 + 4.70i)19-s + (−3.34 + 3.12i)21-s − 7.95·23-s − 1.62·25-s + (−2.37 + 4.62i)27-s + (−0.325 + 0.563i)29-s + (−0.518 + 0.897i)31-s + ⋯
L(s)  = 1  + (−0.934 + 0.357i)3-s − 0.821·5-s + (0.926 − 0.376i)7-s + (0.744 − 0.667i)9-s − 0.933·11-s + (0.666 + 1.15i)13-s + (0.767 − 0.293i)15-s + (0.453 + 0.786i)17-s + (−0.622 + 1.07i)19-s + (−0.730 + 0.682i)21-s − 1.65·23-s − 0.325·25-s + (−0.457 + 0.889i)27-s + (−0.0604 + 0.104i)29-s + (−0.0930 + 0.161i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.463 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.463 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.463 - 0.885i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.463 - 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.320427 + 0.529505i\)
\(L(\frac12)\) \(\approx\) \(0.320427 + 0.529505i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 - 0.618i)T \)
7 \( 1 + (-2.45 + 0.997i)T \)
good5 \( 1 + 1.83T + 5T^{2} \)
11 \( 1 + 3.09T + 11T^{2} \)
13 \( 1 + (-2.40 - 4.16i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.87 - 3.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.71 - 4.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.95T + 23T^{2} \)
29 \( 1 + (0.325 - 0.563i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.518 - 0.897i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.873 + 1.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.52 - 4.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.09 - 10.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.30 - 3.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.55 - 7.88i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.89 + 5.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.40 - 4.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.23 + 12.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.00T + 71T^{2} \)
73 \( 1 + (1.81 + 3.14i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.17 - 12.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.83 + 6.63i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.76 - 9.99i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.04 - 1.80i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.15802010030888724856253577708, −10.57351433396258152035982035365, −9.680783997198326587678595958880, −8.214744170307766006532128393210, −7.80129698377084011353291458527, −6.48148580278007594342790449569, −5.60611085430074953113059550705, −4.35714027326936226227865015420, −3.88610904450418843465798233853, −1.62382476273831838064890964676, 0.42388828408319346369868155269, 2.28623360039012140350760092306, 3.97844602026364338385471487055, 5.15182718838101166945380241948, 5.73044217839708347188587659769, 7.10335421755294469453182550188, 7.908802893204333202922080394845, 8.476582399663437834274576556766, 10.10017341430325691851702154267, 10.77813155422566367307485783162

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