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Properties

Label 2-2736-19.18-c0-0-1
Degree $2$
Conductor $2736$
Sign $1$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s − 11-s + 17-s − 19-s + 2·23-s + 35-s + 43-s − 47-s − 55-s − 61-s − 73-s − 77-s + 2·83-s + 85-s − 95-s − 2·101-s + 2·115-s + 119-s + ⋯
L(s)  = 1  + 5-s + 7-s − 11-s + 17-s − 19-s + 2·23-s + 35-s + 43-s − 47-s − 55-s − 61-s − 73-s − 77-s + 2·83-s + 85-s − 95-s − 2·101-s + 2·115-s + 119-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2736} (721, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.599728238\)
\(L(\frac12)\) \(\approx\) \(1.599728238\)
\(L(1)\) 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 \( 1 - T + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 - T + T^{2} \)
23 \( ( 1 - T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 - T + T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−9.043120676639309277434999195440, −8.198844554149815866554541180374, −7.62219979381158856777276810352, −6.70306557813243787913478861963, −5.77452295403062336094919108895, −5.18789380477369845738117220005, −4.51935419621189112379970413878, −3.17034571443940164985824061281, −2.28253428548503722999720628950, −1.32394939357817066302335234154, 1.32394939357817066302335234154, 2.28253428548503722999720628950, 3.17034571443940164985824061281, 4.51935419621189112379970413878, 5.18789380477369845738117220005, 5.77452295403062336094919108895, 6.70306557813243787913478861963, 7.62219979381158856777276810352, 8.198844554149815866554541180374, 9.043120676639309277434999195440

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