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Properties

Label 2-528-12.11-c5-0-47
Degree $2$
Conductor $528$
Sign $0.785 - 0.618i$
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  + (2.23 + 15.4i)3-s − 55.6i·5-s + 112. i·7-s + (−233. + 68.8i)9-s − 121·11-s + 758.·13-s + (857. − 124. i)15-s − 1.65e3i·17-s + 1.28e3i·19-s + (−1.73e3 + 250. i)21-s − 741.·23-s + 32.6·25-s + (−1.58e3 − 3.44e3i)27-s − 7.26e3i·29-s + 5.22e3i·31-s + ⋯
L(s)  = 1  + (0.143 + 0.989i)3-s − 0.994i·5-s + 0.866i·7-s + (−0.958 + 0.283i)9-s − 0.301·11-s + 1.24·13-s + (0.984 − 0.142i)15-s − 1.38i·17-s + 0.819i·19-s + (−0.857 + 0.124i)21-s − 0.292·23-s + 0.0104·25-s + (−0.417 − 0.908i)27-s − 1.60i·29-s + 0.976i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 - 0.618i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.785 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $0.785 - 0.618i$
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: \((2,\ 528,\ (\ :5/2),\ 0.785 - 0.618i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.166789432\)
\(L(\frac12)\) \(\approx\) \(2.166789432\)
\(L(\frac{7}{2})\) 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 \( 1 + (-2.23 - 15.4i)T \)
11 \( 1 + 121T \)
good5 \( 1 + 55.6iT - 3.12e3T^{2} \)
7 \( 1 - 112. iT - 1.68e4T^{2} \)
13 \( 1 - 758.T + 3.71e5T^{2} \)
17 \( 1 + 1.65e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.28e3iT - 2.47e6T^{2} \)
23 \( 1 + 741.T + 6.43e6T^{2} \)
29 \( 1 + 7.26e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.22e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.41e3T + 6.93e7T^{2} \)
41 \( 1 + 1.45e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.51e4iT - 1.47e8T^{2} \)
47 \( 1 - 9.93e3T + 2.29e8T^{2} \)
53 \( 1 - 2.69e4iT - 4.18e8T^{2} \)
59 \( 1 - 7.19e3T + 7.14e8T^{2} \)
61 \( 1 - 1.20e4T + 8.44e8T^{2} \)
67 \( 1 + 6.25e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.37e4T + 1.80e9T^{2} \)
73 \( 1 - 5.86e4T + 2.07e9T^{2} \)
79 \( 1 - 1.59e4iT - 3.07e9T^{2} \)
83 \( 1 + 7.65e4T + 3.93e9T^{2} \)
89 \( 1 - 2.53e3iT - 5.58e9T^{2} \)
97 \( 1 - 1.56e5T + 8.58e9T^{2} \)
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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

−10.00927547704494419994510989234, −9.135821492101597809219254898731, −8.669863435286348339997588422520, −7.80678530801544341352710699458, −6.12245678181247465901969995450, −5.38941239196414993033248546792, −4.56701233710442530890946747118, −3.49506107081332721750264975442, −2.30518778084871979686452937100, −0.74544952283435601126015768410, 0.72157626759051627574232200588, 1.84737548528676127737996607221, 3.08201725210091589272836266971, 3.94600711727793355430879397071, 5.62389220263541642613696113531, 6.58986748364184025328964773047, 7.08413937441304597218356578090, 8.086946642560019238152141380221, 8.818687229478064382747617299715, 10.24831106628537505972625639190

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