Properties

Label 2-2736-1.1-c1-0-32
Degree $2$
Conductor $2736$
Sign $-1$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
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  − 2·5-s + 2·13-s − 2·17-s + 19-s − 25-s − 2·29-s + 4·31-s + 2·37-s − 6·41-s + 4·43-s − 7·49-s − 10·53-s − 4·59-s − 2·61-s − 4·65-s + 12·67-s − 6·73-s + 4·79-s − 8·83-s + 4·85-s − 6·89-s − 2·95-s − 14·97-s + 6·101-s − 4·103-s − 20·107-s − 6·109-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.554·13-s − 0.485·17-s + 0.229·19-s − 1/5·25-s − 0.371·29-s + 0.718·31-s + 0.328·37-s − 0.937·41-s + 0.609·43-s − 49-s − 1.37·53-s − 0.520·59-s − 0.256·61-s − 0.496·65-s + 1.46·67-s − 0.702·73-s + 0.450·79-s − 0.878·83-s + 0.433·85-s − 0.635·89-s − 0.205·95-s − 1.42·97-s + 0.597·101-s − 0.394·103-s − 1.93·107-s − 0.574·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
19 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.252030560408320635093882963616, −7.87537518402846875743837031097, −6.92822227886234169606129538581, −6.25452895745643782338001477919, −5.27360891135767646869901082305, −4.37266688783284321332139300895, −3.68712589588092002834188844808, −2.75741804606621082589758524445, −1.44248876304714278885152632746, 0, 1.44248876304714278885152632746, 2.75741804606621082589758524445, 3.68712589588092002834188844808, 4.37266688783284321332139300895, 5.27360891135767646869901082305, 6.25452895745643782338001477919, 6.92822227886234169606129538581, 7.87537518402846875743837031097, 8.252030560408320635093882963616

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