Properties

Label 2-12e3-1.1-c1-0-1
Degree $2$
Conductor $1728$
Sign $1$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.60·5-s − 3.60·7-s + 11-s − 4·13-s − 7.21·19-s + 6·23-s + 7.99·25-s + 7.21·29-s + 3.60·31-s + 12.9·35-s − 10·37-s − 7.21·41-s + 7.21·43-s + 10·47-s + 5.99·49-s + 3.60·53-s − 3.60·55-s − 4·59-s + 14.4·65-s + 7.21·67-s − 8·71-s − 3·73-s − 3.60·77-s + 14.4·79-s + 9·83-s + 7.21·89-s + 14.4·91-s + ⋯
L(s)  = 1  − 1.61·5-s − 1.36·7-s + 0.301·11-s − 1.10·13-s − 1.65·19-s + 1.25·23-s + 1.59·25-s + 1.33·29-s + 0.647·31-s + 2.19·35-s − 1.64·37-s − 1.12·41-s + 1.09·43-s + 1.45·47-s + 0.857·49-s + 0.495·53-s − 0.486·55-s − 0.520·59-s + 1.78·65-s + 0.880·67-s − 0.949·71-s − 0.351·73-s − 0.410·77-s + 1.62·79-s + 0.987·83-s + 0.764·89-s + 1.51·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6309043613\)
\(L(\frac12)\) \(\approx\) \(0.6309043613\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.60T + 5T^{2} \)
7 \( 1 + 3.60T + 7T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 7.21T + 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 7.21T + 29T^{2} \)
31 \( 1 - 3.60T + 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + 7.21T + 41T^{2} \)
43 \( 1 - 7.21T + 43T^{2} \)
47 \( 1 - 10T + 47T^{2} \)
53 \( 1 - 3.60T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 7.21T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 3T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 - 7.21T + 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−9.121121629782389058986131123389, −8.618081729078947053173652392082, −7.66326356957408072686519371675, −6.89208770407464967596666447518, −6.45140615756740678946649068375, −5.02545959388557553075024709866, −4.22502165299621719669060191330, −3.42673669077228629319182351147, −2.57465764992036494543346087747, −0.51788370474579765160855112023, 0.51788370474579765160855112023, 2.57465764992036494543346087747, 3.42673669077228629319182351147, 4.22502165299621719669060191330, 5.02545959388557553075024709866, 6.45140615756740678946649068375, 6.89208770407464967596666447518, 7.66326356957408072686519371675, 8.618081729078947053173652392082, 9.121121629782389058986131123389

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