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Properties

Label 2-528-44.43-c5-0-51
Degree $2$
Conductor $528$
Sign $-0.732 + 0.680i$
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 + 1.95·5-s + 158.·7-s − 81·9-s + (−391. + 89.4i)11-s − 135. i·13-s + 17.5i·15-s + 602. i·17-s + 456.·19-s + 1.42e3i·21-s − 3.40e3i·23-s − 3.12e3·25-s − 729i·27-s − 3.32e3i·29-s + 3.73e3i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.0349·5-s + 1.22·7-s − 0.333·9-s + (−0.974 + 0.222i)11-s − 0.222i·13-s + 0.0201i·15-s + 0.505i·17-s + 0.290·19-s + 0.704i·21-s − 1.34i·23-s − 0.998·25-s − 0.192i·27-s − 0.733i·29-s + 0.697i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $-0.732 + 0.680i$
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: \((2,\ 528,\ (\ :5/2),\ -0.732 + 0.680i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.2033817186\)
\(L(\frac12)\) \(\approx\) \(0.2033817186\)
\(L(\frac{7}{2})\) 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 \( 1 + (391. - 89.4i)T \)
good5 \( 1 - 1.95T + 3.12e3T^{2} \)
7 \( 1 - 158.T + 1.68e4T^{2} \)
13 \( 1 + 135. iT - 3.71e5T^{2} \)
17 \( 1 - 602. iT - 1.41e6T^{2} \)
19 \( 1 - 456.T + 2.47e6T^{2} \)
23 \( 1 + 3.40e3iT - 6.43e6T^{2} \)
29 \( 1 + 3.32e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.73e3iT - 2.86e7T^{2} \)
37 \( 1 + 9.30e3T + 6.93e7T^{2} \)
41 \( 1 - 1.39e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.98e4T + 1.47e8T^{2} \)
47 \( 1 + 1.39e4iT - 2.29e8T^{2} \)
53 \( 1 - 3.41e4T + 4.18e8T^{2} \)
59 \( 1 - 2.47e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.05e4iT - 8.44e8T^{2} \)
67 \( 1 - 7.11e4iT - 1.35e9T^{2} \)
71 \( 1 + 7.00e4iT - 1.80e9T^{2} \)
73 \( 1 + 3.13e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.42e4T + 3.07e9T^{2} \)
83 \( 1 + 7.40e4T + 3.93e9T^{2} \)
89 \( 1 - 8.21e3T + 5.58e9T^{2} \)
97 \( 1 + 1.13e5T + 8.58e9T^{2} \)
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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

−10.04376525723375850559687436835, −8.585714453454269066546223280482, −8.193322666995585162175450680792, −7.11438000742175772993938654737, −5.76098268277059034719880615587, −4.97330207558976707708582106848, −4.14207866083020665981244644936, −2.77848728676210502454165094945, −1.63417343068689734865991572438, −0.04317083771701902288986616751, 1.36754367792635091593916671690, 2.27929171065048347905154989410, 3.61880249524034431805514715671, 5.05163516546108472920711504914, 5.58756210694315579497880071823, 6.98870582128222620097509103968, 7.74639096505500304563463093121, 8.389842484720653746035307355716, 9.461779823732701811246709547010, 10.52732660536329784119159272659

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