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Properties

Label 2-528-33.2-c1-0-20
Degree $2$
Conductor $528$
Sign $-0.00778 + 0.999i$
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  + (0.297 − 1.70i)3-s + (2.90 − 0.945i)5-s + (2.14 − 2.95i)7-s + (−2.82 − 1.01i)9-s + (−3.28 − 0.421i)11-s + (1.02 + 0.333i)13-s + (−0.747 − 5.24i)15-s + (0.380 + 1.17i)17-s + (3.04 + 4.19i)19-s + (−4.40 − 4.53i)21-s + 6.43i·23-s + (3.52 − 2.56i)25-s + (−2.57 + 4.51i)27-s + (−6.00 − 4.36i)29-s + (−1.21 + 3.72i)31-s + ⋯
L(s)  = 1  + (0.171 − 0.985i)3-s + (1.30 − 0.422i)5-s + (0.810 − 1.11i)7-s + (−0.941 − 0.338i)9-s + (−0.991 − 0.127i)11-s + (0.285 + 0.0926i)13-s + (−0.193 − 1.35i)15-s + (0.0923 + 0.284i)17-s + (0.698 + 0.961i)19-s + (−0.960 − 0.990i)21-s + 1.34i·23-s + (0.704 − 0.512i)25-s + (−0.494 + 0.868i)27-s + (−1.11 − 0.809i)29-s + (−0.217 + 0.669i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00778 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00778 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $-0.00778 + 0.999i$
Analytic conductor: \(4.21610\)
Root analytic conductor: \(2.05331\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{528} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 528,\ (\ :1/2),\ -0.00778 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32645 - 1.33682i\)
\(L(\frac12)\) \(\approx\) \(1.32645 - 1.33682i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.297 + 1.70i)T \)
11 \( 1 + (3.28 + 0.421i)T \)
good5 \( 1 + (-2.90 + 0.945i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-2.14 + 2.95i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.02 - 0.333i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.380 - 1.17i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.04 - 4.19i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 6.43iT - 23T^{2} \)
29 \( 1 + (6.00 + 4.36i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.21 - 3.72i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.171 + 0.124i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-5.20 + 3.77i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 11.6iT - 43T^{2} \)
47 \( 1 + (-1.87 - 2.57i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-7.80 - 2.53i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.66 + 5.04i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.80 - 1.88i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 3.24T + 67T^{2} \)
71 \( 1 + (-0.315 + 0.102i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.00 - 11.0i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.26 + 0.410i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.41 - 7.42i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 10.3iT - 89T^{2} \)
97 \( 1 + (-1.96 + 6.04i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.61172841577308567424490852676, −9.775133811955557458728660426770, −8.752755181150298920123175210382, −7.76311690322225303418200402005, −7.23440683586067148689256872850, −5.80050924424512254156917189403, −5.38922657653578371319617581850, −3.73252467530313761288603081216, −2.11866864719527678776081056042, −1.20984120322950661451071095049, 2.22308651942711498055499880190, 2.91704491844580606783659739702, 4.71471520612224125311183798058, 5.41149714833348844657752421245, 6.12228410637333257407042237839, 7.63444930857576181652511152870, 8.720013389165096992667551191817, 9.322075950242920341013543676179, 10.16832370309013651311319248917, 10.92071174257340900334221452942

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