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Properties

Label 2-528-33.32-c3-0-36
Degree $2$
Conductor $528$
Sign $0.677 + 0.735i$
Analytic cond. $31.1530$
Root an. cond. $5.58148$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.832 − 5.12i)3-s + 6.09i·5-s + 1.16i·7-s + (−25.6 + 8.54i)9-s + (30.4 − 20.0i)11-s + 37.5i·13-s + (31.2 − 5.07i)15-s − 40.8·17-s − 84.3i·19-s + (6 − 0.973i)21-s + 101. i·23-s + 87.7·25-s + (65.1 + 124. i)27-s + 251.·29-s − 19.9·31-s + ⋯
L(s)  = 1  + (−0.160 − 0.987i)3-s + 0.545i·5-s + 0.0631i·7-s + (−0.948 + 0.316i)9-s + (0.834 − 0.550i)11-s + 0.801i·13-s + (0.538 − 0.0874i)15-s − 0.583·17-s − 1.01i·19-s + (0.0623 − 0.0101i)21-s + 0.917i·23-s + 0.702·25-s + (0.464 + 0.885i)27-s + 1.61·29-s − 0.115·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $0.677 + 0.735i$
Analytic conductor: \(31.1530\)
Root analytic conductor: \(5.58148\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{528} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 528,\ (\ :3/2),\ 0.677 + 0.735i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.795303579\)
\(L(\frac12)\) \(\approx\) \(1.795303579\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.832 + 5.12i)T \)
11 \( 1 + (-30.4 + 20.0i)T \)
good5 \( 1 - 6.09iT - 125T^{2} \)
7 \( 1 - 1.16iT - 343T^{2} \)
13 \( 1 - 37.5iT - 2.19e3T^{2} \)
17 \( 1 + 40.8T + 4.91e3T^{2} \)
19 \( 1 + 84.3iT - 6.85e3T^{2} \)
23 \( 1 - 101. iT - 1.21e4T^{2} \)
29 \( 1 - 251.T + 2.43e4T^{2} \)
31 \( 1 + 19.9T + 2.97e4T^{2} \)
37 \( 1 - 45.9T + 5.06e4T^{2} \)
41 \( 1 - 175.T + 6.89e4T^{2} \)
43 \( 1 + 332. iT - 7.95e4T^{2} \)
47 \( 1 + 186. iT - 1.03e5T^{2} \)
53 \( 1 - 554. iT - 1.48e5T^{2} \)
59 \( 1 + 185. iT - 2.05e5T^{2} \)
61 \( 1 + 754. iT - 2.26e5T^{2} \)
67 \( 1 - 381.T + 3.00e5T^{2} \)
71 \( 1 - 401. iT - 3.57e5T^{2} \)
73 \( 1 + 1.03e3iT - 3.89e5T^{2} \)
79 \( 1 + 1.04e3iT - 4.93e5T^{2} \)
83 \( 1 + 847.T + 5.71e5T^{2} \)
89 \( 1 + 675. iT - 7.04e5T^{2} \)
97 \( 1 + 378.T + 9.12e5T^{2} \)
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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

−10.57930007382042523473655961129, −9.170793628332851044220494348248, −8.624110542028990867652923282170, −7.35300044085221125961225324040, −6.72741743537265833318179175083, −6.01042737278408815456783264628, −4.68318645704223091928938680589, −3.25435489649875626818945989979, −2.12554722748825734783020924070, −0.77430074811755968283667402670, 0.922404366223222147684099599659, 2.73354495428806864837331714260, 4.05587411430549938981826959301, 4.72592521993938282703751828077, 5.79640267622573867040098299926, 6.76294970263568932509224677881, 8.197781752169603941239558676887, 8.816369777338215693822202499966, 9.790640729555926137096261835791, 10.40247421478263601357046913459

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