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Properties

Label 2-528-44.43-c5-0-7
Degree $2$
Conductor $528$
Sign $-0.163 - 0.986i$
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 − 76.1·5-s − 138.·7-s − 81·9-s + (141. − 375. i)11-s − 693. i·13-s − 685. i·15-s + 902. i·17-s − 951.·19-s − 1.24e3i·21-s − 980. i·23-s + 2.67e3·25-s − 729i·27-s − 8.53e3i·29-s − 3.90e3i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.36·5-s − 1.06·7-s − 0.333·9-s + (0.351 − 0.936i)11-s − 1.13i·13-s − 0.786i·15-s + 0.757i·17-s − 0.604·19-s − 0.616i·21-s − 0.386i·23-s + 0.857·25-s − 0.192i·27-s − 1.88i·29-s − 0.729i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $-0.163 - 0.986i$
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: \((2,\ 528,\ (\ :5/2),\ -0.163 - 0.986i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.3248336408\)
\(L(\frac12)\) \(\approx\) \(0.3248336408\)
\(L(\frac{7}{2})\) 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 \( 1 + (-141. + 375. i)T \)
good5 \( 1 + 76.1T + 3.12e3T^{2} \)
7 \( 1 + 138.T + 1.68e4T^{2} \)
13 \( 1 + 693. iT - 3.71e5T^{2} \)
17 \( 1 - 902. iT - 1.41e6T^{2} \)
19 \( 1 + 951.T + 2.47e6T^{2} \)
23 \( 1 + 980. iT - 6.43e6T^{2} \)
29 \( 1 + 8.53e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.90e3iT - 2.86e7T^{2} \)
37 \( 1 - 8.48e3T + 6.93e7T^{2} \)
41 \( 1 - 5.33e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.82e4T + 1.47e8T^{2} \)
47 \( 1 - 254. iT - 2.29e8T^{2} \)
53 \( 1 + 1.40e4T + 4.18e8T^{2} \)
59 \( 1 - 2.65e3iT - 7.14e8T^{2} \)
61 \( 1 - 9.69e3iT - 8.44e8T^{2} \)
67 \( 1 - 3.52e4iT - 1.35e9T^{2} \)
71 \( 1 - 3.30e4iT - 1.80e9T^{2} \)
73 \( 1 + 4.41e4iT - 2.07e9T^{2} \)
79 \( 1 - 7.94e4T + 3.07e9T^{2} \)
83 \( 1 + 1.40e4T + 3.93e9T^{2} \)
89 \( 1 + 1.23e5T + 5.58e9T^{2} \)
97 \( 1 + 3.46e4T + 8.58e9T^{2} \)
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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

−10.32061862688040951742391782775, −9.505618333777090357337918117181, −8.312684368402842875383750327045, −7.958306910589366240552613251917, −6.53014974448276587285361195054, −5.79749896351400835280055155551, −4.32682662807774728534980359503, −3.65385012010562679224557875295, −2.81674435253804341785005533331, −0.59931034944920399891300598672, 0.13040595265219167785392434268, 1.60546473657183996541653905812, 3.04298035455807666343678592432, 3.96744664532812516033038322481, 4.98151626643325547366404129382, 6.66975820006809821048495746766, 6.90210424421662687956209030163, 7.88458258224074807869822832726, 8.940414341291767002322984283796, 9.645748036952998980454910478485

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