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Properties

Label 2-504-56.37-c1-0-19
Degree $2$
Conductor $504$
Sign $0.874 - 0.485i$
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  + (−0.268 + 1.38i)2-s + (−1.85 − 0.744i)4-s + (1.23 + 0.710i)5-s + (1.39 − 2.24i)7-s + (1.53 − 2.37i)8-s + (−1.31 + 1.51i)10-s + (0.832 − 0.480i)11-s − 3.57i·13-s + (2.74 + 2.54i)14-s + (2.89 + 2.76i)16-s + (−2.43 − 4.20i)17-s + (6.28 + 3.62i)19-s + (−1.75 − 2.23i)20-s + (0.444 + 1.28i)22-s + (2.72 − 4.71i)23-s + ⋯
L(s)  = 1  + (−0.189 + 0.981i)2-s + (−0.928 − 0.372i)4-s + (0.550 + 0.317i)5-s + (0.527 − 0.849i)7-s + (0.541 − 0.840i)8-s + (−0.416 + 0.480i)10-s + (0.251 − 0.144i)11-s − 0.990i·13-s + (0.734 + 0.678i)14-s + (0.722 + 0.690i)16-s + (−0.589 − 1.02i)17-s + (1.44 + 0.831i)19-s + (−0.392 − 0.499i)20-s + (0.0947 + 0.273i)22-s + (0.567 − 0.982i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.874 - 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34193 + 0.347332i\)
\(L(\frac12)\) \(\approx\) \(1.34193 + 0.347332i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.268 - 1.38i)T \)
3 \( 1 \)
7 \( 1 + (-1.39 + 2.24i)T \)
good5 \( 1 + (-1.23 - 0.710i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.832 + 0.480i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.57iT - 13T^{2} \)
17 \( 1 + (2.43 + 4.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.28 - 3.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.72 + 4.71i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.78iT - 29T^{2} \)
31 \( 1 + (-3.67 - 6.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.21 - 1.27i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.20T + 41T^{2} \)
43 \( 1 - 4.45iT - 43T^{2} \)
47 \( 1 + (0.211 - 0.366i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.41 + 4.85i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.43 - 3.71i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.67 - 0.969i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.13 - 5.27i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.12T + 71T^{2} \)
73 \( 1 + (4.99 + 8.65i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.139 + 0.241i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.69iT - 83T^{2} \)
89 \( 1 + (-1.07 + 1.86i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.44T + 97T^{2} \)
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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

−10.57447756038346483147721822982, −10.14267867831596656640321174664, −9.096182856508493042890257311257, −8.148661120128596808893492828302, −7.30050330757351427799182745933, −6.54972097952307167247930655474, −5.42326735946763571636170720395, −4.62277757184880380559119424861, −3.18622351004957635477747670883, −1.06854126094673675233710529156, 1.50313722181639596029805258947, 2.49339983366144979735878907760, 3.98666562220761154829438388940, 5.02489949313465975704096365331, 5.94735129273891122211350785033, 7.44640370524571149022201194494, 8.518975808275616898691933393301, 9.331751326402471960683468753180, 9.712311881010481009544640549593, 11.12297119529956976981103908243

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