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Properties

Label 2-504-56.3-c1-0-36
Degree $2$
Conductor $504$
Sign $-0.974 - 0.222i$
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  + (0.725 − 1.21i)2-s + (−0.947 − 1.76i)4-s + (−1.14 − 1.97i)5-s + (−1.95 + 1.78i)7-s + (−2.82 − 0.127i)8-s + (−3.22 − 0.0485i)10-s + (2.60 − 4.50i)11-s − 1.44·13-s + (0.752 + 3.66i)14-s + (−2.20 + 3.33i)16-s + (−1.71 − 0.992i)17-s + (−4.27 + 2.47i)19-s + (−2.39 + 3.88i)20-s + (−3.58 − 6.42i)22-s + (−6.02 + 3.47i)23-s + ⋯
L(s)  = 1  + (0.512 − 0.858i)2-s + (−0.473 − 0.880i)4-s + (−0.510 − 0.883i)5-s + (−0.737 + 0.675i)7-s + (−0.998 − 0.0451i)8-s + (−1.02 − 0.0153i)10-s + (0.784 − 1.35i)11-s − 0.400·13-s + (0.201 + 0.979i)14-s + (−0.551 + 0.834i)16-s + (−0.416 − 0.240i)17-s + (−0.981 + 0.566i)19-s + (−0.536 + 0.867i)20-s + (−0.763 − 1.36i)22-s + (−1.25 + 0.725i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.974 - 0.222i$
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: \((2,\ 504,\ (\ :1/2),\ -0.974 - 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.107248 + 0.951297i\)
\(L(\frac12)\) \(\approx\) \(0.107248 + 0.951297i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.725 + 1.21i)T \)
3 \( 1 \)
7 \( 1 + (1.95 - 1.78i)T \)
good5 \( 1 + (1.14 + 1.97i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.60 + 4.50i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.44T + 13T^{2} \)
17 \( 1 + (1.71 + 0.992i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.27 - 2.47i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.02 - 3.47i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.21iT - 29T^{2} \)
31 \( 1 + (-1.69 + 2.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.53 + 1.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.20iT - 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + (-1.25 - 2.17i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.15 + 1.82i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.59 - 4.96i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.57 + 6.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.73 + 4.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.6iT - 71T^{2} \)
73 \( 1 + (9.13 + 5.27i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.38 + 2.53i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.265iT - 83T^{2} \)
89 \( 1 + (8.23 - 4.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.7iT - 97T^{2} \)
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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

−10.59800889771350386919264566885, −9.482855723462855241042876598586, −8.936236924167593496540982712159, −8.045386132322009485247487940002, −6.25776132164514605907361981254, −5.75195387860204033075477137280, −4.38229994190648281612136714875, −3.63092854237610981883809342510, −2.26563071031776542317099292982, −0.46875153263253193138363532864, 2.66713037332178165687511797806, 3.96405252750989621285242926902, 4.52136872800778765029954344021, 6.17904815916161841426340921476, 6.92846651974662400654829512079, 7.31227164453038990531830841813, 8.530521118768870702406028001540, 9.592368718405530234036303136694, 10.44427695023009627142652252124, 11.51561492815821500682978438789

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