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Properties

Label 2-12e3-12.11-c1-0-9
Degree $2$
Conductor $1728$
Sign $-i$
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  + 5.19i·7-s + 5·13-s − 5.19i·19-s + 5·25-s + 10.3i·31-s − 11·37-s + 10.3i·43-s − 20·49-s + 61-s + 15.5i·67-s + 7·73-s + 5.19i·79-s + 25.9i·91-s + 19·97-s − 15.5i·103-s + ⋯
L(s)  = 1  + 1.96i·7-s + 1.38·13-s − 1.19i·19-s + 25-s + 1.86i·31-s − 1.80·37-s + 1.58i·43-s − 2.85·49-s + 0.128·61-s + 1.90i·67-s + 0.819·73-s + 0.584i·79-s + 2.72i·91-s + 1.92·97-s − 1.53i·103-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-i$
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: \((2,\ 1728,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.644042605\)
\(L(\frac12)\) \(\approx\) \(1.644042605\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T^{2} \)
7 \( 1 - 5.19iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 + 11T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - T + 61T^{2} \)
67 \( 1 - 15.5iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 - 5.19iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 19T + 97T^{2} \)
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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

−9.179117645641228085374461407739, −8.761492191281057470668963627567, −8.311768040010242561740227027625, −6.93870982405224431352355528219, −6.30209165720759638904061657559, −5.42498258955546119253794807754, −4.81737287889976349914890574103, −3.37196803332135349648728696540, −2.65376227532883829966412341319, −1.44693117457659556241888629086, 0.66548116290545091716965141568, 1.74627801398433593820702445156, 3.55866893260058756196291698298, 3.82844166399980370741968220918, 4.90170804215799989605392620031, 6.04119973376921399667922473197, 6.77023319392183657572526330163, 7.56089527598095856668717856638, 8.206281514278240616285755181634, 9.112223179559070144782893456098

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