Properties

Label 2-12e3-4.3-c2-0-41
Degree $2$
Conductor $1728$
Sign $1$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.22·5-s + 6.57i·7-s − 13.8i·11-s + 19.0·13-s + 23.0·17-s + 18.8i·19-s − 13.5i·23-s − 14.5·25-s − 0.752·29-s − 46.4i·31-s + 21.1i·35-s + 2.68·37-s − 34.1·41-s + 20.9i·43-s + 15.4i·47-s + ⋯
L(s)  = 1  + 0.645·5-s + 0.938i·7-s − 1.26i·11-s + 1.46·13-s + 1.35·17-s + 0.991i·19-s − 0.591i·23-s − 0.583·25-s − 0.0259·29-s − 1.49i·31-s + 0.605i·35-s + 0.0724·37-s − 0.832·41-s + 0.486i·43-s + 0.327i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.622884156\)
\(L(\frac12)\) \(\approx\) \(2.622884156\)
\(L(2)\) 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 \( 1 - 3.22T + 25T^{2} \)
7 \( 1 - 6.57iT - 49T^{2} \)
11 \( 1 + 13.8iT - 121T^{2} \)
13 \( 1 - 19.0T + 169T^{2} \)
17 \( 1 - 23.0T + 289T^{2} \)
19 \( 1 - 18.8iT - 361T^{2} \)
23 \( 1 + 13.5iT - 529T^{2} \)
29 \( 1 + 0.752T + 841T^{2} \)
31 \( 1 + 46.4iT - 961T^{2} \)
37 \( 1 - 2.68T + 1.36e3T^{2} \)
41 \( 1 + 34.1T + 1.68e3T^{2} \)
43 \( 1 - 20.9iT - 1.84e3T^{2} \)
47 \( 1 - 15.4iT - 2.20e3T^{2} \)
53 \( 1 - 46.8T + 2.80e3T^{2} \)
59 \( 1 + 40.4iT - 3.48e3T^{2} \)
61 \( 1 - 105.T + 3.72e3T^{2} \)
67 \( 1 - 27.9iT - 4.48e3T^{2} \)
71 \( 1 + 24.0iT - 5.04e3T^{2} \)
73 \( 1 + 120.T + 5.32e3T^{2} \)
79 \( 1 + 95.6iT - 6.24e3T^{2} \)
83 \( 1 - 115. iT - 6.88e3T^{2} \)
89 \( 1 - 169.T + 7.92e3T^{2} \)
97 \( 1 + 93.1T + 9.40e3T^{2} \)
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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

−9.050708969033842079936862523576, −8.385696222598398024817914297515, −7.82407718805822463333476029701, −6.36754620065328886246644342415, −5.85383860535186867872473527473, −5.46259332509171071946703896832, −3.92756588715366879696389115088, −3.18003661423528055504375555462, −2.03671156391229720273808163512, −0.908135406409234137233289337103, 0.988435413613614603708136115045, 1.86830006350095114456484087911, 3.26473832329013798222392096743, 4.06926109197254529311945892143, 5.09424491280942415794320141542, 5.87178827999754965947634884051, 6.91152285145126010103021990712, 7.35376643737461280990565793686, 8.397551191950960898691847069455, 9.184690591718623962730765471229

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