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Properties

Label 2-12e2-1.1-c1-0-0
Degree $2$
Conductor $144$
Sign $1$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
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  + 4·7-s + 2·13-s − 8·19-s − 5·25-s + 4·31-s − 10·37-s − 8·43-s + 9·49-s + 14·61-s + 16·67-s − 10·73-s + 4·79-s + 8·91-s + 14·97-s − 20·103-s + 2·109-s + ⋯
L(s)  = 1  + 1.51·7-s + 0.554·13-s − 1.83·19-s − 25-s + 0.718·31-s − 1.64·37-s − 1.21·43-s + 9/7·49-s + 1.79·61-s + 1.95·67-s − 1.17·73-s + 0.450·79-s + 0.838·91-s + 1.42·97-s − 1.97·103-s + 0.191·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.214325323\)
\(L(\frac12)\) \(\approx\) \(1.214325323\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.16759329932274946491561130296, −11.94687294246884746693957837802, −11.12315991811672295662035767470, −10.22525649903538281666764318697, −8.662053039688615672125838925077, −8.061333242868226986239179539676, −6.62696738834082263668854358565, −5.24060664699283394077045448716, −4.05061428132231144121252913355, −1.92209901273574427656973400179, 1.92209901273574427656973400179, 4.05061428132231144121252913355, 5.24060664699283394077045448716, 6.62696738834082263668854358565, 8.061333242868226986239179539676, 8.662053039688615672125838925077, 10.22525649903538281666764318697, 11.12315991811672295662035767470, 11.94687294246884746693957837802, 13.16759329932274946491561130296

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