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Properties

Label 2-504-56.53-c1-0-2
Degree $2$
Conductor $504$
Sign $-0.960 + 0.279i$
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  + (−0.264 + 1.38i)2-s + (−1.85 − 0.735i)4-s + (0.937 − 0.541i)5-s + (−1.62 + 2.09i)7-s + (1.51 − 2.38i)8-s + (0.503 + 1.44i)10-s + (−4.52 − 2.61i)11-s + 4.43i·13-s + (−2.47 − 2.80i)14-s + (2.91 + 2.73i)16-s + (−3.65 + 6.33i)17-s + (−2.42 + 1.39i)19-s + (−2.14 + 0.317i)20-s + (4.82 − 5.59i)22-s + (−1.51 − 2.62i)23-s + ⋯
L(s)  = 1  + (−0.187 + 0.982i)2-s + (−0.929 − 0.367i)4-s + (0.419 − 0.242i)5-s + (−0.612 + 0.790i)7-s + (0.535 − 0.844i)8-s + (0.159 + 0.457i)10-s + (−1.36 − 0.787i)11-s + 1.23i·13-s + (−0.661 − 0.749i)14-s + (0.729 + 0.684i)16-s + (−0.886 + 1.53i)17-s + (−0.555 + 0.320i)19-s + (−0.478 + 0.0708i)20-s + (1.02 − 1.19i)22-s + (−0.315 − 0.546i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.960 + 0.279i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.960 + 0.279i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0700315 - 0.491818i\)
\(L(\frac12)\) \(\approx\) \(0.0700315 - 0.491818i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.264 - 1.38i)T \)
3 \( 1 \)
7 \( 1 + (1.62 - 2.09i)T \)
good5 \( 1 + (-0.937 + 0.541i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.52 + 2.61i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.43iT - 13T^{2} \)
17 \( 1 + (3.65 - 6.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.42 - 1.39i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.51 + 2.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.06iT - 29T^{2} \)
31 \( 1 + (-3.20 + 5.55i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.85 - 4.53i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.05T + 41T^{2} \)
43 \( 1 - 10.7iT - 43T^{2} \)
47 \( 1 + (4.54 + 7.86i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.71 - 0.989i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.01 + 3.47i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.85 + 3.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.26 - 3.03i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.53T + 71T^{2} \)
73 \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.621 - 1.07i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.1iT - 83T^{2} \)
89 \( 1 + (-3.02 - 5.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.82T + 97T^{2} \)
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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

−11.25715681366769607239060473663, −10.23633716824363768526063800862, −9.453264102569975261071294707602, −8.552183410618342787459963153719, −8.040551362447020909864878326216, −6.50573817459119257716275768663, −6.09895656959967359748624241502, −5.09090632985803176563367894425, −3.88866300613498981587750604560, −2.14328762822979588925389050060, 0.29600818430014018537028518613, 2.30848346336240223024397189531, 3.17820981991045292748556730566, 4.55474908534203797643758728275, 5.44528520766824178400434666710, 6.99362370638359444416334341144, 7.75830414343693562487393636852, 8.908096240160255859223340925612, 9.917243695603542567236605733098, 10.38744294474137071055985464340

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