Properties

Label 2-2736-1.1-c1-0-6
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·11-s + 2·13-s + 6·17-s + 19-s − 4·23-s − 25-s + 2·29-s − 4·31-s + 10·37-s − 10·41-s − 4·43-s − 4·47-s − 7·49-s + 10·53-s + 8·55-s + 12·59-s + 14·61-s − 4·65-s + 12·67-s + 8·71-s − 6·73-s + 4·79-s + 12·83-s − 12·85-s + 6·89-s − 2·95-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.229·19-s − 0.834·23-s − 1/5·25-s + 0.371·29-s − 0.718·31-s + 1.64·37-s − 1.56·41-s − 0.609·43-s − 0.583·47-s − 49-s + 1.37·53-s + 1.07·55-s + 1.56·59-s + 1.79·61-s − 0.496·65-s + 1.46·67-s + 0.949·71-s − 0.702·73-s + 0.450·79-s + 1.31·83-s − 1.30·85-s + 0.635·89-s − 0.205·95-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: $\chi_{2736} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.281924272\)
\(L(\frac12)\) \(\approx\) \(1.281924272\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−8.501995439933935386642992901059, −8.057224083383321383561004935430, −7.54188937600423423101808663065, −6.59739773494044819386161896201, −5.60441811878021212455097225817, −5.02382748416547556621669886282, −3.86301997176052042051145556459, −3.33052988139656335920774502455, −2.15606001082909168122252072981, −0.69408070740984200675842868720, 0.69408070740984200675842868720, 2.15606001082909168122252072981, 3.33052988139656335920774502455, 3.86301997176052042051145556459, 5.02382748416547556621669886282, 5.60441811878021212455097225817, 6.59739773494044819386161896201, 7.54188937600423423101808663065, 8.057224083383321383561004935430, 8.501995439933935386642992901059

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