Properties

Label 2-528-1.1-c3-0-27
Degree $2$
Conductor $528$
Sign $-1$
Analytic cond. $31.1530$
Root an. cond. $5.58148$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 2·7-s + 9·9-s + 11·11-s − 88·13-s − 66·17-s + 40·19-s − 6·21-s − 6·23-s − 125·25-s + 27·27-s − 54·29-s − 8·31-s + 33·33-s − 106·37-s − 264·39-s + 354·41-s + 124·43-s − 546·47-s − 339·49-s − 198·51-s − 408·53-s + 120·57-s − 552·59-s + 404·61-s − 18·63-s + 4·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.107·7-s + 1/3·9-s + 0.301·11-s − 1.87·13-s − 0.941·17-s + 0.482·19-s − 0.0623·21-s − 0.0543·23-s − 25-s + 0.192·27-s − 0.345·29-s − 0.0463·31-s + 0.174·33-s − 0.470·37-s − 1.08·39-s + 1.34·41-s + 0.439·43-s − 1.69·47-s − 0.988·49-s − 0.543·51-s − 1.05·53-s + 0.278·57-s − 1.21·59-s + 0.847·61-s − 0.0359·63-s + 0.00729·67-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: yes
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 \( 1 - p T \)
11 \( 1 - p T \)
good5 \( 1 + p^{3} T^{2} \)
7 \( 1 + 2 T + p^{3} T^{2} \)
13 \( 1 + 88 T + p^{3} T^{2} \)
17 \( 1 + 66 T + p^{3} T^{2} \)
19 \( 1 - 40 T + p^{3} T^{2} \)
23 \( 1 + 6 T + p^{3} T^{2} \)
29 \( 1 + 54 T + p^{3} T^{2} \)
31 \( 1 + 8 T + p^{3} T^{2} \)
37 \( 1 + 106 T + p^{3} T^{2} \)
41 \( 1 - 354 T + p^{3} T^{2} \)
43 \( 1 - 124 T + p^{3} T^{2} \)
47 \( 1 + 546 T + p^{3} T^{2} \)
53 \( 1 + 408 T + p^{3} T^{2} \)
59 \( 1 + 552 T + p^{3} T^{2} \)
61 \( 1 - 404 T + p^{3} T^{2} \)
67 \( 1 - 4 T + p^{3} T^{2} \)
71 \( 1 + 126 T + p^{3} T^{2} \)
73 \( 1 + 166 T + p^{3} T^{2} \)
79 \( 1 - 874 T + p^{3} T^{2} \)
83 \( 1 + 444 T + p^{3} T^{2} \)
89 \( 1 - 1002 T + p^{3} T^{2} \)
97 \( 1 + 802 T + p^{3} T^{2} \)
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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

−9.630665918183156262110895605482, −9.418703316272549288760296782226, −8.112570125036595800721534811804, −7.39348593850653414503716623376, −6.46645444213010471277424024054, −5.14527282862758253208693712306, −4.20100356018503928093528298325, −2.92765916361044637269055357121, −1.88319409568422524194315749441, 0, 1.88319409568422524194315749441, 2.92765916361044637269055357121, 4.20100356018503928093528298325, 5.14527282862758253208693712306, 6.46645444213010471277424024054, 7.39348593850653414503716623376, 8.112570125036595800721534811804, 9.418703316272549288760296782226, 9.630665918183156262110895605482

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