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Properties

Label 2-528-12.11-c5-0-73
Degree $2$
Conductor $528$
Sign $0.999 + 0.0150i$
Analytic cond. $84.6826$
Root an. cond. $9.20231$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (15.5 + 0.234i)3-s + 48.8i·5-s + 18.7i·7-s + (242. + 7.30i)9-s − 121·11-s + 855.·13-s + (−11.4 + 760. i)15-s − 1.25e3i·17-s − 3.08e3i·19-s + (−4.39 + 292. i)21-s + 427.·23-s + 742.·25-s + (3.78e3 + 170. i)27-s − 6.31e3i·29-s + 7.45e3i·31-s + ⋯
L(s)  = 1  + (0.999 + 0.0150i)3-s + 0.873i·5-s + 0.144i·7-s + (0.999 + 0.0300i)9-s − 0.301·11-s + 1.40·13-s + (−0.0131 + 0.873i)15-s − 1.05i·17-s − 1.96i·19-s + (−0.00217 + 0.144i)21-s + 0.168·23-s + 0.237·25-s + (0.998 + 0.0450i)27-s − 1.39i·29-s + 1.39i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0150i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $0.999 + 0.0150i$
Analytic conductor: \(84.6826\)
Root analytic conductor: \(9.20231\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{528} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 528,\ (\ :5/2),\ 0.999 + 0.0150i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.632938886\)
\(L(\frac12)\) \(\approx\) \(3.632938886\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-15.5 - 0.234i)T \)
11 \( 1 + 121T \)
good5 \( 1 - 48.8iT - 3.12e3T^{2} \)
7 \( 1 - 18.7iT - 1.68e4T^{2} \)
13 \( 1 - 855.T + 3.71e5T^{2} \)
17 \( 1 + 1.25e3iT - 1.41e6T^{2} \)
19 \( 1 + 3.08e3iT - 2.47e6T^{2} \)
23 \( 1 - 427.T + 6.43e6T^{2} \)
29 \( 1 + 6.31e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.45e3iT - 2.86e7T^{2} \)
37 \( 1 + 9.12e3T + 6.93e7T^{2} \)
41 \( 1 + 7.08e3iT - 1.15e8T^{2} \)
43 \( 1 + 4.15e3iT - 1.47e8T^{2} \)
47 \( 1 - 4.66e3T + 2.29e8T^{2} \)
53 \( 1 - 1.67e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.71e4T + 7.14e8T^{2} \)
61 \( 1 - 4.03e4T + 8.44e8T^{2} \)
67 \( 1 - 4.60e4iT - 1.35e9T^{2} \)
71 \( 1 + 9.58e3T + 1.80e9T^{2} \)
73 \( 1 + 3.03e4T + 2.07e9T^{2} \)
79 \( 1 - 4.68e4iT - 3.07e9T^{2} \)
83 \( 1 + 2.63e4T + 3.93e9T^{2} \)
89 \( 1 - 1.19e5iT - 5.58e9T^{2} \)
97 \( 1 - 4.24e4T + 8.58e9T^{2} \)
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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

−10.05665127060342794755822838362, −8.964602283205587888190190903652, −8.507011639527001180717802424898, −7.15564048196317770985631859187, −6.83256702651598820075406641010, −5.36200108292157867167301802696, −4.11640972781919246074383005373, −3.04458382397777691332265043980, −2.39571506269903245473040647552, −0.848683862132479721982153754788, 1.08439701558161451348606522800, 1.86836834041369889547391146108, 3.47148024865292270758215317515, 4.06209801629080495061003936308, 5.37435557094719506652871377628, 6.41328288380763317106230395565, 7.69238258956696655117662467250, 8.437880308333715652549116323777, 8.874353687367074567968752176188, 10.03736760399877542128670347820

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