Properties

Label 2-288-1.1-c1-0-3
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 6·13-s − 8·17-s + 11·25-s + 4·29-s − 2·37-s + 8·41-s − 7·49-s + 4·53-s − 10·61-s + 24·65-s + 6·73-s + 32·85-s − 16·89-s − 18·97-s + 20·101-s − 6·109-s − 16·113-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.66·13-s − 1.94·17-s + 11/5·25-s + 0.742·29-s − 0.328·37-s + 1.24·41-s − 49-s + 0.549·53-s − 1.28·61-s + 2.97·65-s + 0.702·73-s + 3.47·85-s − 1.69·89-s − 1.82·97-s + 1.99·101-s − 0.574·109-s − 1.50·113-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: $\chi_{288} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
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 \( 1 + 16 T + p T^{2} \)
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

−11.43133431305815887793598810757, −10.64173856953551402265625207636, −9.327959901425306425198768202784, −8.338442505726918333995265218224, −7.46428835432640754113355987046, −6.69932041253260706274722644089, −4.85517883035181554193637871803, −4.15127486396668832984271170028, −2.68900975784905443020785166460, 0, 2.68900975784905443020785166460, 4.15127486396668832984271170028, 4.85517883035181554193637871803, 6.69932041253260706274722644089, 7.46428835432640754113355987046, 8.338442505726918333995265218224, 9.327959901425306425198768202784, 10.64173856953551402265625207636, 11.43133431305815887793598810757

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