Properties

Label 2-528-1.1-c3-0-28
Degree $2$
Conductor $528$
Sign $-1$
Analytic cond. $31.1530$
Root an. cond. $5.58148$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 1.12·5-s − 0.876·7-s + 9·9-s − 11·11-s − 71.0·13-s + 3.36·15-s + 22.2·17-s − 125.·19-s − 2.63·21-s + 104.·23-s − 123.·25-s + 27·27-s − 303.·29-s + 196.·31-s − 33·33-s − 0.984·35-s + 247.·37-s − 213.·39-s − 448.·41-s + 196.·43-s + 10.1·45-s − 28.5·47-s − 342.·49-s + 66.6·51-s − 38.4·53-s − 12.3·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.100·5-s − 0.0473·7-s + 0.333·9-s − 0.301·11-s − 1.51·13-s + 0.0579·15-s + 0.316·17-s − 1.51·19-s − 0.0273·21-s + 0.948·23-s − 0.989·25-s + 0.192·27-s − 1.94·29-s + 1.14·31-s − 0.174·33-s − 0.00475·35-s + 1.10·37-s − 0.875·39-s − 1.70·41-s + 0.696·43-s + 0.0334·45-s − 0.0885·47-s − 0.997·49-s + 0.182·51-s − 0.0996·53-s − 0.0302·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $-1$
Analytic conductor: \(31.1530\)
Root analytic conductor: \(5.58148\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 528,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 3T \)
11 \( 1 + 11T \)
good5 \( 1 - 1.12T + 125T^{2} \)
7 \( 1 + 0.876T + 343T^{2} \)
13 \( 1 + 71.0T + 2.19e3T^{2} \)
17 \( 1 - 22.2T + 4.91e3T^{2} \)
19 \( 1 + 125.T + 6.85e3T^{2} \)
23 \( 1 - 104.T + 1.21e4T^{2} \)
29 \( 1 + 303.T + 2.43e4T^{2} \)
31 \( 1 - 196.T + 2.97e4T^{2} \)
37 \( 1 - 247.T + 5.06e4T^{2} \)
41 \( 1 + 448.T + 6.89e4T^{2} \)
43 \( 1 - 196.T + 7.95e4T^{2} \)
47 \( 1 + 28.5T + 1.03e5T^{2} \)
53 \( 1 + 38.4T + 1.48e5T^{2} \)
59 \( 1 + 14.8T + 2.05e5T^{2} \)
61 \( 1 + 625.T + 2.26e5T^{2} \)
67 \( 1 + 668.T + 3.00e5T^{2} \)
71 \( 1 - 179.T + 3.57e5T^{2} \)
73 \( 1 - 1.05e3T + 3.89e5T^{2} \)
79 \( 1 + 458.T + 4.93e5T^{2} \)
83 \( 1 - 626.T + 5.71e5T^{2} \)
89 \( 1 + 1.57e3T + 7.04e5T^{2} \)
97 \( 1 - 827.T + 9.12e5T^{2} \)
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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

−9.860855441564568517159637555797, −9.231084084779624122174398597821, −8.125075571621738972200171035291, −7.44068169875648795437332552054, −6.40491472909925589863170417806, −5.18315595222357040805036811438, −4.20051098005224296151715928499, −2.89979216271398834782547103014, −1.90350896958212712454798310565, 0, 1.90350896958212712454798310565, 2.89979216271398834782547103014, 4.20051098005224296151715928499, 5.18315595222357040805036811438, 6.40491472909925589863170417806, 7.44068169875648795437332552054, 8.125075571621738972200171035291, 9.231084084779624122174398597821, 9.860855441564568517159637555797

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