Properties

Label 2-12e3-1.1-c3-0-59
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 yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 11.4·5-s + 29.8·7-s + 66.2·11-s − 39.8·13-s + 107.·17-s + 70.3·19-s − 6.91·23-s + 5.33·25-s − 36.6·29-s + 231.·31-s + 340.·35-s − 36.8·37-s + 429.·41-s − 74.3·43-s − 52.5·47-s + 546.·49-s − 288.·53-s + 756.·55-s − 783.·59-s − 439.·61-s − 454.·65-s − 218.·67-s − 790.·71-s + 1.09e3·73-s + 1.97e3·77-s − 439.·79-s − 50.8·83-s + ⋯
L(s)  = 1  + 1.02·5-s + 1.61·7-s + 1.81·11-s − 0.849·13-s + 1.53·17-s + 0.849·19-s − 0.0627·23-s + 0.0426·25-s − 0.234·29-s + 1.34·31-s + 1.64·35-s − 0.163·37-s + 1.63·41-s − 0.263·43-s − 0.163·47-s + 1.59·49-s − 0.746·53-s + 1.85·55-s − 1.72·59-s − 0.921·61-s − 0.867·65-s − 0.398·67-s − 1.32·71-s + 1.76·73-s + 2.92·77-s − 0.626·79-s − 0.0672·83-s + ⋯

Functional equation

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

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: $\chi_{1728} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.290071831\)
\(L(\frac12)\) \(\approx\) \(4.290071831\)
\(L(\frac{5}{2})\) 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 \( 1 - 11.4T + 125T^{2} \)
7 \( 1 - 29.8T + 343T^{2} \)
11 \( 1 - 66.2T + 1.33e3T^{2} \)
13 \( 1 + 39.8T + 2.19e3T^{2} \)
17 \( 1 - 107.T + 4.91e3T^{2} \)
19 \( 1 - 70.3T + 6.85e3T^{2} \)
23 \( 1 + 6.91T + 1.21e4T^{2} \)
29 \( 1 + 36.6T + 2.43e4T^{2} \)
31 \( 1 - 231.T + 2.97e4T^{2} \)
37 \( 1 + 36.8T + 5.06e4T^{2} \)
41 \( 1 - 429.T + 6.89e4T^{2} \)
43 \( 1 + 74.3T + 7.95e4T^{2} \)
47 \( 1 + 52.5T + 1.03e5T^{2} \)
53 \( 1 + 288.T + 1.48e5T^{2} \)
59 \( 1 + 783.T + 2.05e5T^{2} \)
61 \( 1 + 439.T + 2.26e5T^{2} \)
67 \( 1 + 218.T + 3.00e5T^{2} \)
71 \( 1 + 790.T + 3.57e5T^{2} \)
73 \( 1 - 1.09e3T + 3.89e5T^{2} \)
79 \( 1 + 439.T + 4.93e5T^{2} \)
83 \( 1 + 50.8T + 5.71e5T^{2} \)
89 \( 1 - 719.T + 7.04e5T^{2} \)
97 \( 1 + 1.19e3T + 9.12e5T^{2} \)
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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

−9.210981706177144223936553175816, −8.037734608372937347590407404934, −7.53888082995470534991793904964, −6.45449177217074427172400006373, −5.67554976613937624703579401705, −4.92729727852737634473357027064, −4.09013391643110489201823776612, −2.82290041078521775750871177476, −1.59435355266943973355241035071, −1.18119744194327315900281415741, 1.18119744194327315900281415741, 1.59435355266943973355241035071, 2.82290041078521775750871177476, 4.09013391643110489201823776612, 4.92729727852737634473357027064, 5.67554976613937624703579401705, 6.45449177217074427172400006373, 7.53888082995470534991793904964, 8.037734608372937347590407404934, 9.210981706177144223936553175816

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