Properties

Label 2-12e2-1.1-c3-0-6
Degree $2$
Conductor $144$
Sign $-1$
Analytic cond. $8.49627$
Root an. cond. $2.91483$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 24·7-s − 44·11-s + 22·13-s − 50·17-s − 44·19-s − 56·23-s − 121·25-s − 198·29-s + 160·31-s − 48·35-s − 162·37-s + 198·41-s − 52·43-s + 528·47-s + 233·49-s + 242·53-s − 88·55-s − 668·59-s + 550·61-s + 44·65-s − 188·67-s + 728·71-s + 154·73-s + 1.05e3·77-s + 656·79-s + 236·83-s + ⋯
L(s)  = 1  + 0.178·5-s − 1.29·7-s − 1.20·11-s + 0.469·13-s − 0.713·17-s − 0.531·19-s − 0.507·23-s − 0.967·25-s − 1.26·29-s + 0.926·31-s − 0.231·35-s − 0.719·37-s + 0.754·41-s − 0.184·43-s + 1.63·47-s + 0.679·49-s + 0.627·53-s − 0.215·55-s − 1.47·59-s + 1.15·61-s + 0.0839·65-s − 0.342·67-s + 1.21·71-s + 0.246·73-s + 1.56·77-s + 0.934·79-s + 0.312·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(8.49627\)
Root analytic conductor: \(2.91483\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{144} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 144,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 2 T + p^{3} T^{2} \)
7 \( 1 + 24 T + p^{3} T^{2} \)
11 \( 1 + 4 p T + p^{3} T^{2} \)
13 \( 1 - 22 T + p^{3} T^{2} \)
17 \( 1 + 50 T + p^{3} T^{2} \)
19 \( 1 + 44 T + p^{3} T^{2} \)
23 \( 1 + 56 T + p^{3} T^{2} \)
29 \( 1 + 198 T + p^{3} T^{2} \)
31 \( 1 - 160 T + p^{3} T^{2} \)
37 \( 1 + 162 T + p^{3} T^{2} \)
41 \( 1 - 198 T + p^{3} T^{2} \)
43 \( 1 + 52 T + p^{3} T^{2} \)
47 \( 1 - 528 T + p^{3} T^{2} \)
53 \( 1 - 242 T + p^{3} T^{2} \)
59 \( 1 + 668 T + p^{3} T^{2} \)
61 \( 1 - 550 T + p^{3} T^{2} \)
67 \( 1 + 188 T + p^{3} T^{2} \)
71 \( 1 - 728 T + p^{3} T^{2} \)
73 \( 1 - 154 T + p^{3} T^{2} \)
79 \( 1 - 656 T + p^{3} T^{2} \)
83 \( 1 - 236 T + p^{3} T^{2} \)
89 \( 1 + 714 T + p^{3} T^{2} \)
97 \( 1 + 478 T + p^{3} T^{2} \)
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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

−12.32760643063906801292048222691, −10.95567051856758266008967035182, −10.07889937262292264094866961725, −9.107953158667345285609407307501, −7.86091764770730364802195417256, −6.57954820252895661457513691372, −5.58999877781664199789035588774, −3.90571484976180381229755600963, −2.45130381914706935386949996740, 0, 2.45130381914706935386949996740, 3.90571484976180381229755600963, 5.58999877781664199789035588774, 6.57954820252895661457513691372, 7.86091764770730364802195417256, 9.107953158667345285609407307501, 10.07889937262292264094866961725, 10.95567051856758266008967035182, 12.32760643063906801292048222691

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