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Properties

Label 2-528-44.43-c5-0-52
Degree $2$
Conductor $528$
Sign $-0.784 + 0.620i$
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  − 9i·3-s + 48.1·5-s − 32.4·7-s − 81·9-s + (397. − 58.0i)11-s − 711. i·13-s − 433. i·15-s + 528. i·17-s − 1.61e3·19-s + 292. i·21-s − 2.48e3i·23-s − 805.·25-s + 729i·27-s − 3.53e3i·29-s + 9.75e3i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.861·5-s − 0.250·7-s − 0.333·9-s + (0.989 − 0.144i)11-s − 1.16i·13-s − 0.497i·15-s + 0.443i·17-s − 1.02·19-s + 0.144i·21-s − 0.979i·23-s − 0.257·25-s + 0.192i·27-s − 0.781i·29-s + 1.82i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $-0.784 + 0.620i$
Analytic conductor: \(84.6826\)
Root analytic conductor: \(9.20231\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{528} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 528,\ (\ :5/2),\ -0.784 + 0.620i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.706351964\)
\(L(\frac12)\) \(\approx\) \(1.706351964\)
\(L(\frac{7}{2})\) 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 \( 1 + 9iT \)
11 \( 1 + (-397. + 58.0i)T \)
good5 \( 1 - 48.1T + 3.12e3T^{2} \)
7 \( 1 + 32.4T + 1.68e4T^{2} \)
13 \( 1 + 711. iT - 3.71e5T^{2} \)
17 \( 1 - 528. iT - 1.41e6T^{2} \)
19 \( 1 + 1.61e3T + 2.47e6T^{2} \)
23 \( 1 + 2.48e3iT - 6.43e6T^{2} \)
29 \( 1 + 3.53e3iT - 2.05e7T^{2} \)
31 \( 1 - 9.75e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.39e4T + 6.93e7T^{2} \)
41 \( 1 + 1.11e4iT - 1.15e8T^{2} \)
43 \( 1 + 9.93e3T + 1.47e8T^{2} \)
47 \( 1 + 1.29e4iT - 2.29e8T^{2} \)
53 \( 1 - 9.90e3T + 4.18e8T^{2} \)
59 \( 1 + 2.79e4iT - 7.14e8T^{2} \)
61 \( 1 + 5.36e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.19e4iT - 1.35e9T^{2} \)
71 \( 1 - 7.36e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.46e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.78e4T + 3.07e9T^{2} \)
83 \( 1 - 1.04e5T + 3.93e9T^{2} \)
89 \( 1 - 5.98e4T + 5.58e9T^{2} \)
97 \( 1 + 9.83e4T + 8.58e9T^{2} \)
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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

−9.782794933516453508090340000076, −8.726547125030500023055300408146, −8.046877591167386065525927858720, −6.68331345698284302899707309985, −6.23920547418704182271665135474, −5.22227884016034307565397728346, −3.83550074170724238702034048536, −2.59827804404991891776147541891, −1.56077158403186921508559635024, −0.36830177853549287498391890991, 1.38862218742983267386321857343, 2.47076388520103143925976872949, 3.86265765362201359957490654665, 4.65394486733449339032957581852, 5.96336353252400449752360273832, 6.50345387283625857880910578884, 7.72809215515322343776018762173, 9.145756396297243766798446042624, 9.359587369358078225250725687042, 10.20768695960880426835983424457

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