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Properties

Label 2-504-8.5-c1-0-24
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

\(L(1)\) \(\approx\) \(2.46172 - 0.738764i\)
\(L(\frac12)\) \(\approx\) \(2.46172 - 0.738764i\)
\(L(\frac{3}{2})\) 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

−11.22803793248443917896389370170, −9.881297893777932655914963031661, −8.931977480919542701917975067700, −8.133312578586084746205825722158, −7.05680628770973378274082061201, −5.88413170108088253429187578379, −5.15155548572226560793892234269, −4.26089284551487750275491854703, −3.09393308268596948614226618836, −1.32295708074804636269149931935, 2.08556022607243273494987235065, 3.25426856340053004424912898820, 3.89659246598650247075497730684, 5.65331564530764215055282589642, 6.09010603347294436429470159557, 7.40234855219658044859771950313, 7.68623890543358871236035109576, 9.872079916524735755515919316367, 10.20582922638862135345318590443, 10.94958584787958897143899017015

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