Properties

Label 2-12e3-8.5-c1-0-4
Degree $2$
Conductor $1728$
Sign $0.258 - 0.965i$
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  − 3.46i·5-s + 1.73·7-s + 6i·11-s + 5.19i·13-s − 6·17-s + 5i·19-s − 3.46·23-s − 6.99·25-s + 6.92i·29-s + 3.46·31-s − 5.99i·35-s + 1.73i·37-s + 4i·43-s − 3.46·47-s − 4·49-s + ⋯
L(s)  = 1  − 1.54i·5-s + 0.654·7-s + 1.80i·11-s + 1.44i·13-s − 1.45·17-s + 1.14i·19-s − 0.722·23-s − 1.39·25-s + 1.28i·29-s + 0.622·31-s − 1.01i·35-s + 0.284i·37-s + 0.609i·43-s − 0.505·47-s − 0.571·49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.258 - 0.965i$
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: \((2,\ 1728,\ (\ :1/2),\ 0.258 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.260652062\)
\(L(\frac12)\) \(\approx\) \(1.260652062\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.46iT - 5T^{2} \)
7 \( 1 - 1.73T + 7T^{2} \)
11 \( 1 - 6iT - 11T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 5iT - 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 - 1.73iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + 6.92iT - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 - 12.1iT - 61T^{2} \)
67 \( 1 + 5iT - 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 - 5.19T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
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

−9.446430848658213181077014742951, −8.683592967123184188202908398842, −8.096771643236108218939553033330, −7.12737328493612465233984636695, −6.33732903377607424362326405054, −5.03126809022214819815043317543, −4.62871576144711991545922893177, −4.02122885964939440890393802578, −1.99546764743876528837745447358, −1.57728332918893853765682201915, 0.45898129849462273206057108039, 2.37757415549403651165607299319, 3.00835853466204950898612402687, 3.95127181805860418111835907324, 5.17335353397650782263019465923, 6.13634146391230126780042335600, 6.58968893445860970865590411012, 7.70500906430223323505978353636, 8.205369183937142223744885601834, 9.072083358310088914124931237469

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