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Properties

Label 2-504-56.37-c1-0-8
Degree $2$
Conductor $504$
Sign $-0.861 - 0.508i$
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  + (0.535 + 1.30i)2-s + (−1.42 + 1.40i)4-s + (1.93 + 1.11i)5-s + (−0.5 + 2.59i)7-s + (−2.59 − 1.11i)8-s + (−0.427 + 3.13i)10-s + (−1.93 + 1.11i)11-s + (−3.66 + 0.736i)14-s + (0.0729 − 3.99i)16-s + (−1.73 − 3i)17-s + (6.70 + 3.87i)19-s + (−4.33 + 1.11i)20-s + (−2.5 − 1.93i)22-s + (−3.46 + 6i)23-s + (−2.92 − 4.40i)28-s − 2.23i·29-s + ⋯
L(s)  = 1  + (0.378 + 0.925i)2-s + (−0.713 + 0.700i)4-s + (0.866 + 0.499i)5-s + (−0.188 + 0.981i)7-s + (−0.918 − 0.395i)8-s + (−0.135 + 0.990i)10-s + (−0.583 + 0.337i)11-s + (−0.980 + 0.196i)14-s + (0.0182 − 0.999i)16-s + (−0.420 − 0.727i)17-s + (1.53 + 0.888i)19-s + (−0.968 + 0.249i)20-s + (−0.533 − 0.412i)22-s + (−0.722 + 1.25i)23-s + (−0.553 − 0.833i)28-s − 0.415i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.861 - 0.508i$
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: \((2,\ 504,\ (\ :1/2),\ -0.861 - 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.421078 + 1.54205i\)
\(L(\frac12)\) \(\approx\) \(0.421078 + 1.54205i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.535 - 1.30i)T \)
3 \( 1 \)
7 \( 1 + (0.5 - 2.59i)T \)
good5 \( 1 + (-1.93 - 1.11i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.93 - 1.11i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (1.73 + 3i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.70 - 3.87i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.46 - 6i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.23iT - 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.70 - 3.87i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-1.73 + 3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.68 + 5.59i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.93 - 1.11i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.70 - 3.87i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.70 + 3.87i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 + (-6.92 + 12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−11.59292707046403171196011717429, −9.835443887414448712751088206241, −9.715409403612406784169162350769, −8.449865742013433732875234401979, −7.55964772803863570979632594914, −6.57414275012425718318760190358, −5.67197959941844404176912266807, −5.12201698972351506730799431026, −3.49259034821350737947299631432, −2.34887529411121416229912199421, 0.870525629130244860981422230326, 2.31698603892674907360740716663, 3.61202545785461006335084909666, 4.74123243799109605452595686867, 5.59745952460715091570793988223, 6.65669700915017025723877570860, 8.041758019660777874970877968272, 9.102556415216809830108465886783, 9.860647187683105020953347442738, 10.55533568318195079208417204378

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