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Properties

Label 2-528-11.3-c1-0-6
Degree $2$
Conductor $528$
Sign $0.927 - 0.374i$
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  + (0.309 + 0.951i)3-s + (−1.30 − 0.951i)5-s + (0.309 − 0.951i)7-s + (−0.809 + 0.587i)9-s + (2.19 + 2.48i)11-s + (3.42 − 2.48i)13-s + (0.499 − 1.53i)15-s + (6.35 + 4.61i)17-s + (0.263 + 0.812i)19-s + 0.999·21-s + 4.23·23-s + (−0.736 − 2.26i)25-s + (−0.809 − 0.587i)27-s + (−1.85 + 5.70i)29-s + (4.11 − 2.99i)31-s + ⋯
L(s)  = 1  + (0.178 + 0.549i)3-s + (−0.585 − 0.425i)5-s + (0.116 − 0.359i)7-s + (−0.269 + 0.195i)9-s + (0.660 + 0.750i)11-s + (0.950 − 0.690i)13-s + (0.129 − 0.397i)15-s + (1.54 + 1.11i)17-s + (0.0605 + 0.186i)19-s + 0.218·21-s + 0.883·23-s + (−0.147 − 0.453i)25-s + (−0.155 − 0.113i)27-s + (−0.344 + 1.05i)29-s + (0.739 − 0.537i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(4.21610\)
Root analytic conductor: \(2.05331\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{528} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 528,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49647 + 0.290668i\)
\(L(\frac12)\) \(\approx\) \(1.49647 + 0.290668i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-2.19 - 2.48i)T \)
good5 \( 1 + (1.30 + 0.951i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.42 + 2.48i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-6.35 - 4.61i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.263 - 0.812i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.23T + 23T^{2} \)
29 \( 1 + (1.85 - 5.70i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.11 + 2.99i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.545 - 1.67i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.30 + 4.02i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.70T + 43T^{2} \)
47 \( 1 + (0.336 + 1.03i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.11 + 1.53i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.97 - 9.14i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.92 + 5.03i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 4.85T + 67T^{2} \)
71 \( 1 + (4.30 + 3.13i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.38 + 7.33i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.89 + 6.46i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.04 + 4.39i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 3.76T + 89T^{2} \)
97 \( 1 + (0.927 - 0.673i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.69340570129320383659503736736, −10.16777323851459658682156166135, −9.076882152023305278653220657384, −8.260110010368480672430007517645, −7.52668654084555782034205695872, −6.24063408749171162125644644539, −5.14107717063392356061389273755, −4.07207484727429997390069937532, −3.33876422215237416204958238618, −1.30670135757995685253229504568, 1.18032925801039025986051188143, 2.92284246713361823344164983185, 3.78630782259760262917383768015, 5.29667354012067147119695017239, 6.35223372298610106966755289166, 7.18234704934809047224821763301, 8.096073878633141762288580516973, 8.907846145620244432749844361360, 9.792629973221128393952241395066, 11.19082007051280508657178462783

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