Properties

Label 2-288-1.1-c1-0-1
Degree $2$
Conductor $288$
Sign $1$
Analytic cond. $2.29969$
Root an. cond. $1.51647$
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 + 6·13-s − 2·17-s − 25-s + 10·29-s − 2·37-s − 10·41-s − 7·49-s − 14·53-s − 10·61-s + 12·65-s − 6·73-s − 4·85-s − 10·89-s + 18·97-s + 2·101-s + 6·109-s + 14·113-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.66·13-s − 0.485·17-s − 1/5·25-s + 1.85·29-s − 0.328·37-s − 1.56·41-s − 49-s − 1.92·53-s − 1.28·61-s + 1.48·65-s − 0.702·73-s − 0.433·85-s − 1.05·89-s + 1.82·97-s + 0.199·101-s + 0.574·109-s + 1.31·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.513845634\)
\(L(\frac12)\) \(\approx\) \(1.513845634\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−11.74104183190432762188729414275, −10.79424580459691548950831956254, −9.988093934240586032989036843847, −8.944850116469648867972974935175, −8.155593376401658883389001410970, −6.62912355829218159331214255175, −5.98788019309426758968327152208, −4.69735185642958092576120623346, −3.23795791772265289886094041940, −1.61821392697617915675557729976, 1.61821392697617915675557729976, 3.23795791772265289886094041940, 4.69735185642958092576120623346, 5.98788019309426758968327152208, 6.62912355829218159331214255175, 8.155593376401658883389001410970, 8.944850116469648867972974935175, 9.988093934240586032989036843847, 10.79424580459691548950831956254, 11.74104183190432762188729414275

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