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Properties

Label 2-12e3-12.11-c3-0-66
Degree $2$
Conductor $1728$
Sign $1$
Analytic cond. $101.955$
Root an. cond. $10.0972$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.49i·5-s + 26.1i·7-s + 56.3·11-s + 41.3·13-s − 51.0i·17-s − 79.0i·19-s − 27.3·23-s + 122.·25-s − 134. i·29-s + 187. i·31-s − 39.1·35-s + 196.·37-s − 298. i·41-s − 465. i·43-s + 373.·47-s + ⋯
L(s)  = 1  + 0.133i·5-s + 1.41i·7-s + 1.54·11-s + 0.881·13-s − 0.728i·17-s − 0.954i·19-s − 0.248·23-s + 0.982·25-s − 0.861i·29-s + 1.08i·31-s − 0.189·35-s + 0.873·37-s − 1.13i·41-s − 1.65i·43-s + 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(101.955\)
Root analytic conductor: \(10.0972\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.686986326\)
\(L(\frac12)\) \(\approx\) \(2.686986326\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.49iT - 125T^{2} \)
7 \( 1 - 26.1iT - 343T^{2} \)
11 \( 1 - 56.3T + 1.33e3T^{2} \)
13 \( 1 - 41.3T + 2.19e3T^{2} \)
17 \( 1 + 51.0iT - 4.91e3T^{2} \)
19 \( 1 + 79.0iT - 6.85e3T^{2} \)
23 \( 1 + 27.3T + 1.21e4T^{2} \)
29 \( 1 + 134. iT - 2.43e4T^{2} \)
31 \( 1 - 187. iT - 2.97e4T^{2} \)
37 \( 1 - 196.T + 5.06e4T^{2} \)
41 \( 1 + 298. iT - 6.89e4T^{2} \)
43 \( 1 + 465. iT - 7.95e4T^{2} \)
47 \( 1 - 373.T + 1.03e5T^{2} \)
53 \( 1 + 620. iT - 1.48e5T^{2} \)
59 \( 1 + 321.T + 2.05e5T^{2} \)
61 \( 1 + 674.T + 2.26e5T^{2} \)
67 \( 1 - 576. iT - 3.00e5T^{2} \)
71 \( 1 + 223.T + 3.57e5T^{2} \)
73 \( 1 - 70.1T + 3.89e5T^{2} \)
79 \( 1 + 1.05e3iT - 4.93e5T^{2} \)
83 \( 1 - 1.21e3T + 5.71e5T^{2} \)
89 \( 1 - 1.34e3iT - 7.04e5T^{2} \)
97 \( 1 + 576.T + 9.12e5T^{2} \)
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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

−8.978716139739894126847566132611, −8.490963994478124285143249762468, −7.21869213831032235832525617160, −6.51273678388293944746280901880, −5.80911259567354126037077904247, −4.92109037391091411838832713540, −3.88401663536814898358517320705, −2.91072558116383680319985735617, −1.95078739028331705739857220041, −0.72809569125894633211679568591, 0.992774645619208045845945285911, 1.46954412504691762454721301419, 3.21735351305393628735489215741, 4.04446748399460751400120026643, 4.49769011466643112848028518006, 6.06421972977889406224527414717, 6.40134734147361093553425728072, 7.43938667066077852231850774874, 8.085506608014153822974747233930, 9.034005183784751315687987139252

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