Properties

Label 2-12e3-12.11-c1-0-2
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 no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.23i·5-s + 3.87i·7-s − 1.73·11-s − 2·13-s + 4.47i·17-s − 6.92·23-s − 4.47i·29-s − 3.87i·31-s − 8.66·35-s + 4·37-s − 8.94i·41-s + 7.74i·43-s − 3.46·47-s − 8.00·49-s + 2.23i·53-s + ⋯
L(s)  = 1  + 0.999i·5-s + 1.46i·7-s − 0.522·11-s − 0.554·13-s + 1.08i·17-s − 1.44·23-s − 0.830i·29-s − 0.695i·31-s − 1.46·35-s + 0.657·37-s − 1.39i·41-s + 1.18i·43-s − 0.505·47-s − 1.14·49-s + 0.307i·53-s + ⋯

Functional equation

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

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: $\chi_{1728} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8379584553\)
\(L(\frac12)\) \(\approx\) \(0.8379584553\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23iT - 5T^{2} \)
7 \( 1 - 3.87iT - 7T^{2} \)
11 \( 1 + 1.73T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 4.47iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 6.92T + 23T^{2} \)
29 \( 1 + 4.47iT - 29T^{2} \)
31 \( 1 + 3.87iT - 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 8.94iT - 41T^{2} \)
43 \( 1 - 7.74iT - 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 2.23iT - 53T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + 7.74iT - 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 5T + 73T^{2} \)
79 \( 1 + 7.74iT - 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 - 4.47iT - 89T^{2} \)
97 \( 1 - 11T + 97T^{2} \)
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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

−9.806409758342984961893525753998, −8.910505547143542810093234296352, −8.077061751780683620669816171409, −7.46920201193895123316003397862, −6.17047392633193031559096400435, −6.01832541292989651788101354156, −4.87356676384399571926684490045, −3.73021094408857367259475447845, −2.63601843455134035974488438199, −2.08617075506423563662371953794, 0.30718620337990413194323516852, 1.44122967911840906743579244448, 2.89097057270423967486267410496, 4.06792729940957682415109316632, 4.74342490083514527970801202610, 5.45624154831014595108225179134, 6.69866474754708933470649130867, 7.40460137676977333962314835880, 8.074362320036344303505657708236, 8.903924501709111508943579648491

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