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Properties

Label 2-528-11.9-c1-0-8
Degree $2$
Conductor $528$
Sign $0.263 + 0.964i$
Analytic cond. $4.21610$
Root an. cond. $2.05331$
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  + (0.809 − 0.587i)3-s + (−0.809 − 2.48i)5-s + (2.42 + 1.76i)7-s + (0.309 − 0.951i)9-s + (−1.69 − 2.85i)11-s + (0.545 − 1.67i)13-s + (−2.11 − 1.53i)15-s + (0.5 + 1.53i)17-s + (4.73 − 3.44i)19-s + 3·21-s − 3.47·23-s + (−1.49 + 1.08i)25-s + (−0.309 − 0.951i)27-s + (−3.61 − 2.62i)29-s + (−0.881 + 2.71i)31-s + ⋯
L(s)  = 1  + (0.467 − 0.339i)3-s + (−0.361 − 1.11i)5-s + (0.917 + 0.666i)7-s + (0.103 − 0.317i)9-s + (−0.509 − 0.860i)11-s + (0.151 − 0.465i)13-s + (−0.546 − 0.397i)15-s + (0.121 + 0.373i)17-s + (1.08 − 0.789i)19-s + 0.654·21-s − 0.723·23-s + (−0.299 + 0.217i)25-s + (−0.0594 − 0.183i)27-s + (−0.671 − 0.488i)29-s + (−0.158 + 0.487i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $0.263 + 0.964i$
Analytic conductor: \(4.21610\)
Root analytic conductor: \(2.05331\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{528} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 528,\ (\ :1/2),\ 0.263 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31474 - 1.00382i\)
\(L(\frac12)\) \(\approx\) \(1.31474 - 1.00382i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (1.69 + 2.85i)T \)
good5 \( 1 + (0.809 + 2.48i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.42 - 1.76i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.545 + 1.67i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.5 - 1.53i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.73 + 3.44i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.47T + 23T^{2} \)
29 \( 1 + (3.61 + 2.62i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.881 - 2.71i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.190 + 0.138i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-9.66 + 7.02i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 6.23T + 43T^{2} \)
47 \( 1 + (-1.30 + 0.951i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.97 - 9.14i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-8.35 - 6.06i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.42 - 7.46i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 9.56T + 67T^{2} \)
71 \( 1 + (-1.71 - 5.29i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.61 - 1.90i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.92 - 9.00i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.218 + 0.673i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 0.527T + 89T^{2} \)
97 \( 1 + (4.33 - 13.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.81945079722963612147812222714, −9.521406257344114694775739536144, −8.605019097817071956030570618098, −8.234835579236336996756427498326, −7.36148890830955698500590822973, −5.75763102117051697649734095292, −5.14577892442151859101857090510, −3.89224251299620582378531038708, −2.50200943325781693419355793934, −1.00736001656009906182059920955, 1.90386278304849756031703084614, 3.25679001194425205530546324413, 4.21339144885266706588915706933, 5.26459553189847253737609243245, 6.73843283263469067819251960737, 7.61406963808823430139356993047, 8.023240916618786539331224873313, 9.517173080602365905039343995329, 10.12040898731993251269792462431, 11.07004932868320009130670005672

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