Properties

Label 2-528-1.1-c3-0-26
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 + 19.4·5-s − 6.74·7-s + 9·9-s − 11·11-s − 60.9·13-s − 58.4·15-s − 99.1·17-s − 24.7·19-s + 20.2·21-s − 112·23-s + 254.·25-s − 27·27-s − 21.1·29-s + 318.·31-s + 33·33-s − 131.·35-s − 150.·37-s + 182.·39-s − 252.·41-s − 214.·43-s + 175.·45-s − 105.·47-s − 297.·49-s + 297.·51-s + 325.·53-s − 214.·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.74·5-s − 0.364·7-s + 0.333·9-s − 0.301·11-s − 1.30·13-s − 1.00·15-s − 1.41·17-s − 0.298·19-s + 0.210·21-s − 1.01·23-s + 2.03·25-s − 0.192·27-s − 0.135·29-s + 1.84·31-s + 0.174·33-s − 0.634·35-s − 0.668·37-s + 0.751·39-s − 0.962·41-s − 0.759·43-s + 0.581·45-s − 0.328·47-s − 0.867·49-s + 0.816·51-s + 0.843·53-s − 0.525·55-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 \( 1 - 19.4T + 125T^{2} \)
7 \( 1 + 6.74T + 343T^{2} \)
13 \( 1 + 60.9T + 2.19e3T^{2} \)
17 \( 1 + 99.1T + 4.91e3T^{2} \)
19 \( 1 + 24.7T + 6.85e3T^{2} \)
23 \( 1 + 112T + 1.21e4T^{2} \)
29 \( 1 + 21.1T + 2.43e4T^{2} \)
31 \( 1 - 318.T + 2.97e4T^{2} \)
37 \( 1 + 150.T + 5.06e4T^{2} \)
41 \( 1 + 252.T + 6.89e4T^{2} \)
43 \( 1 + 214.T + 7.95e4T^{2} \)
47 \( 1 + 105.T + 1.03e5T^{2} \)
53 \( 1 - 325.T + 1.48e5T^{2} \)
59 \( 1 + 196T + 2.05e5T^{2} \)
61 \( 1 + 402.T + 2.26e5T^{2} \)
67 \( 1 + 27.4T + 3.00e5T^{2} \)
71 \( 1 - 300.T + 3.57e5T^{2} \)
73 \( 1 - 427.T + 3.89e5T^{2} \)
79 \( 1 + 97.5T + 4.93e5T^{2} \)
83 \( 1 + 1.10e3T + 5.71e5T^{2} \)
89 \( 1 - 463.T + 7.04e5T^{2} \)
97 \( 1 + 1.79e3T + 9.12e5T^{2} \)
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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

−10.00975496386647978677834835633, −9.442732485679175872997419683629, −8.321671860512676192372816728977, −6.84959523401718471156734771170, −6.35230872150857586439864271099, −5.35836803531288767170612636688, −4.56082661505276639723512582519, −2.68543577132564273521205376323, −1.79911492739856569575008455635, 0, 1.79911492739856569575008455635, 2.68543577132564273521205376323, 4.56082661505276639723512582519, 5.35836803531288767170612636688, 6.35230872150857586439864271099, 6.84959523401718471156734771170, 8.321671860512676192372816728977, 9.442732485679175872997419683629, 10.00975496386647978677834835633

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