Properties

Label 2-76e2-1.1-c1-0-98
Degree $2$
Conductor $5776$
Sign $-1$
Analytic cond. $46.1215$
Root an. cond. $6.79128$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 2·9-s + 4·11-s + 13-s + 15-s + 3·17-s − 5·23-s − 4·25-s + 5·27-s − 7·29-s + 4·31-s − 4·33-s − 10·37-s − 39-s + 5·41-s + 5·43-s + 2·45-s + 7·47-s − 7·49-s − 3·51-s − 11·53-s − 4·55-s + 3·59-s + 11·61-s − 65-s − 3·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s + 1.20·11-s + 0.277·13-s + 0.258·15-s + 0.727·17-s − 1.04·23-s − 4/5·25-s + 0.962·27-s − 1.29·29-s + 0.718·31-s − 0.696·33-s − 1.64·37-s − 0.160·39-s + 0.780·41-s + 0.762·43-s + 0.298·45-s + 1.02·47-s − 49-s − 0.420·51-s − 1.51·53-s − 0.539·55-s + 0.390·59-s + 1.40·61-s − 0.124·65-s − 0.366·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5776\)    =    \(2^{4} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(46.1215\)
Root analytic conductor: \(6.79128\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5776,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 11 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−7.81064688678413294283936427077, −6.98313030740649667003756277168, −6.20242670333156375507606244660, −5.75037007890491983002414816625, −4.93263465315888049599527407383, −3.89418238021085879834872170716, −3.54128321504213286476163725449, −2.27475654680847836243871132897, −1.18503774075683578904122444259, 0, 1.18503774075683578904122444259, 2.27475654680847836243871132897, 3.54128321504213286476163725449, 3.89418238021085879834872170716, 4.93263465315888049599527407383, 5.75037007890491983002414816625, 6.20242670333156375507606244660, 6.98313030740649667003756277168, 7.81064688678413294283936427077

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