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Properties

Label 2-504-8.5-c1-0-0
Degree $2$
Conductor $504$
Sign $-0.834 - 0.550i$
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  + (−1.38 − 0.273i)2-s + (1.85 + 0.758i)4-s + 3.66i·5-s − 7-s + (−2.36 − 1.55i)8-s + (0.999 − 5.07i)10-s + 2.56i·11-s + 3.03i·13-s + (1.38 + 0.273i)14-s + (2.85 + 2.80i)16-s − 7.49·17-s − 7.12i·19-s + (−2.77 + 6.77i)20-s + (0.701 − 3.56i)22-s + 3.60·23-s + ⋯
L(s)  = 1  + (−0.981 − 0.193i)2-s + (0.925 + 0.379i)4-s + 1.63i·5-s − 0.377·7-s + (−0.834 − 0.550i)8-s + (0.316 − 1.60i)10-s + 0.774i·11-s + 0.840i·13-s + (0.370 + 0.0730i)14-s + (0.712 + 0.701i)16-s − 1.81·17-s − 1.63i·19-s + (−0.620 + 1.51i)20-s + (0.149 − 0.759i)22-s + 0.751·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.834 - 0.550i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.834 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148897 + 0.496158i\)
\(L(\frac12)\) \(\approx\) \(0.148897 + 0.496158i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.273i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 3.66iT - 5T^{2} \)
11 \( 1 - 2.56iT - 11T^{2} \)
13 \( 1 - 3.03iT - 13T^{2} \)
17 \( 1 + 7.49T + 17T^{2} \)
19 \( 1 + 7.12iT - 19T^{2} \)
23 \( 1 - 3.60T + 23T^{2} \)
29 \( 1 + 6.22iT - 29T^{2} \)
31 \( 1 + 5.40T + 31T^{2} \)
37 \( 1 - 7.12iT - 37T^{2} \)
41 \( 1 + 3.60T + 41T^{2} \)
43 \( 1 - 10.1iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 8.41iT - 53T^{2} \)
59 \( 1 + 2.18iT - 59T^{2} \)
61 \( 1 - 3.03iT - 61T^{2} \)
67 \( 1 - 10.1iT - 67T^{2} \)
71 \( 1 - 7.49T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 7.32iT - 83T^{2} \)
89 \( 1 - 3.60T + 89T^{2} \)
97 \( 1 + 8.80T + 97T^{2} \)
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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

−11.29451134541784762112173313990, −10.39107232085172014911957083846, −9.537280316161945397705230979231, −8.855331445856712636324093015358, −7.46158436104429921641701330261, −6.80844125777732196248410387421, −6.40247375550844936994636746758, −4.38955007783649851099693987999, −2.95150187762957956409919668434, −2.18891592047645111544345858535, 0.39925880843306809168005623519, 1.82957672781817417110682895312, 3.59644286571156204596816977262, 5.15454830335186781834318334067, 5.87804310208850796904960542288, 7.07181563576493164194199754470, 8.235793620866807713386800197517, 8.744579467349257871265494940552, 9.371025779627986741438086482337, 10.47235421618668812998433159886

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