Properties

Label 2-2736-1.1-c1-0-37
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  + 2·11-s − 2·13-s − 4·17-s − 19-s + 2·23-s − 5·25-s − 6·29-s − 4·31-s + 2·37-s + 2·41-s − 4·43-s + 10·47-s − 7·49-s − 6·53-s + 4·59-s + 2·61-s − 4·67-s + 4·71-s − 6·73-s − 8·79-s − 2·83-s − 6·89-s − 10·97-s − 4·101-s + 12·107-s − 2·109-s − 6·113-s + ⋯
L(s)  = 1  + 0.603·11-s − 0.554·13-s − 0.970·17-s − 0.229·19-s + 0.417·23-s − 25-s − 1.11·29-s − 0.718·31-s + 0.328·37-s + 0.312·41-s − 0.609·43-s + 1.45·47-s − 49-s − 0.824·53-s + 0.520·59-s + 0.256·61-s − 0.488·67-s + 0.474·71-s − 0.702·73-s − 0.900·79-s − 0.219·83-s − 0.635·89-s − 1.01·97-s − 0.398·101-s + 1.16·107-s − 0.191·109-s − 0.564·113-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 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.520239651037753866461440447793, −7.60820300443946301756507573395, −6.97927636979287199989982732329, −6.16538576759036497927727588232, −5.36480072812578862147204675104, −4.41001558977101132375039434376, −3.71781377304070401695118209864, −2.55478131435143596783436387949, −1.60866031845956994785102159217, 0, 1.60866031845956994785102159217, 2.55478131435143596783436387949, 3.71781377304070401695118209864, 4.41001558977101132375039434376, 5.36480072812578862147204675104, 6.16538576759036497927727588232, 6.97927636979287199989982732329, 7.60820300443946301756507573395, 8.520239651037753866461440447793

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