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Properties

Label 2-504-8.5-c1-0-11
Degree $2$
Conductor $504$
Sign $-0.254 - 0.967i$
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  + (1.28 + 0.599i)2-s + (1.28 + 1.53i)4-s + 3.33i·5-s − 7-s + (0.719 + 2.73i)8-s + (−2 + 4.27i)10-s − 4.27i·11-s + 3.33i·13-s + (−1.28 − 0.599i)14-s + (−0.719 + 3.93i)16-s − 2·17-s − 0.936i·19-s + (−5.12 + 4.27i)20-s + (2.56 − 5.47i)22-s + 3.12·23-s + ⋯
L(s)  = 1  + (0.905 + 0.424i)2-s + (0.640 + 0.768i)4-s + 1.49i·5-s − 0.377·7-s + (0.254 + 0.967i)8-s + (−0.632 + 1.35i)10-s − 1.28i·11-s + 0.924i·13-s + (−0.342 − 0.160i)14-s + (−0.179 + 0.983i)16-s − 0.485·17-s − 0.214i·19-s + (−1.14 + 0.955i)20-s + (0.546 − 1.16i)22-s + 0.651·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.254 - 0.967i$
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: \((2,\ 504,\ (\ :1/2),\ -0.254 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39216 + 1.80551i\)
\(L(\frac12)\) \(\approx\) \(1.39216 + 1.80551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 - 0.599i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 3.33iT - 5T^{2} \)
11 \( 1 + 4.27iT - 11T^{2} \)
13 \( 1 - 3.33iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 0.936iT - 19T^{2} \)
23 \( 1 - 3.12T + 23T^{2} \)
29 \( 1 + 1.87iT - 29T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 + 1.87iT - 37T^{2} \)
41 \( 1 - 12.2T + 41T^{2} \)
43 \( 1 - 4.27iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 8.54iT - 53T^{2} \)
59 \( 1 - 7.60iT - 59T^{2} \)
61 \( 1 - 3.33iT - 61T^{2} \)
67 \( 1 + 15.7iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 9.47iT - 83T^{2} \)
89 \( 1 + 0.246T + 89T^{2} \)
97 \( 1 + 4.24T + 97T^{2} \)
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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

−11.16649364058328434291353823346, −10.73701071316107559500368704956, −9.378631618414725169585139742318, −8.250336559292191818788769523636, −7.18048868992637040371945385627, −6.49992043484608572877308968968, −5.87049560584157886858600442569, −4.38478896978038924125998168348, −3.30546226600309459997193306610, −2.52479129934916017308338430212, 1.10251923814389671908805055877, 2.57132012261913177191749122072, 4.08247276020000443687536122688, 4.84505366950805220155504978134, 5.63887132278701090706981106817, 6.81843562019810327053091460628, 7.916875614428652147566611606294, 9.110797868993729848476632984955, 9.833486023076490026839330658475, 10.72584080830589915332559849921

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