Properties

Label 2-12e3-8.5-c1-0-11
Degree $2$
Conductor $1728$
Sign $0.965 - 0.258i$
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  + 1.73i·5-s − 1.73·7-s − 3i·11-s − 4i·19-s + 6.92·23-s + 2.00·25-s + 6.92i·29-s + 1.73·31-s − 2.99i·35-s − 6.92i·37-s + 12·41-s + 4i·43-s + 6.92·47-s − 4·49-s + 8.66i·53-s + ⋯
L(s)  = 1  + 0.774i·5-s − 0.654·7-s − 0.904i·11-s − 0.917i·19-s + 1.44·23-s + 0.400·25-s + 1.28i·29-s + 0.311·31-s − 0.507i·35-s − 1.13i·37-s + 1.87·41-s + 0.609i·43-s + 1.01·47-s − 0.571·49-s + 1.18i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.965 - 0.258i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.965 - 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.585299325\)
\(L(\frac12)\) \(\approx\) \(1.585299325\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT - 5T^{2} \)
7 \( 1 + 1.73T + 7T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 - 1.73T + 31T^{2} \)
37 \( 1 + 6.92iT - 37T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 - 8.66iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 13.8iT - 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
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

−9.080449595565580801015494499325, −8.908426926887122025096390991486, −7.55205633561207163351075826194, −6.98514393760380709742419250960, −6.22417144782084210461322435579, −5.40647391490685740915476853802, −4.29375335481292585763692248025, −3.13416304321371494793385260856, −2.71213234152231511023630437156, −0.891480011585764825567005429385, 0.866517614864784807904012604643, 2.19542065185260846578651047525, 3.36409475172383386513422954363, 4.40279785624555169438914778532, 5.09884466326590865152101930158, 6.09492444499203225553077473177, 6.87958882679624216702667005372, 7.75999916730005381707199925563, 8.531253322025214680518102583124, 9.421967193597420800004380587091

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