Properties

Label 2-12e3-12.11-c3-0-53
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  + 20.8i·5-s − 13.9i·7-s − 34.5·11-s + 31.3·13-s + 34.4i·17-s − 120. i·19-s + 137.·23-s − 309.·25-s + 93.1i·29-s − 111. i·31-s + 289.·35-s + 146.·37-s − 8.44i·41-s − 427. i·43-s − 318.·47-s + ⋯
L(s)  = 1  + 1.86i·5-s − 0.750i·7-s − 0.945·11-s + 0.668·13-s + 0.491i·17-s − 1.45i·19-s + 1.24·23-s − 2.47·25-s + 0.596i·29-s − 0.645i·31-s + 1.40·35-s + 0.651·37-s − 0.0321i·41-s − 1.51i·43-s − 0.989·47-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

\(L(2)\) \(\approx\) \(1.827853808\)
\(L(\frac12)\) \(\approx\) \(1.827853808\)
\(L(\frac{5}{2})\) 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 \( 1 - 20.8iT - 125T^{2} \)
7 \( 1 + 13.9iT - 343T^{2} \)
11 \( 1 + 34.5T + 1.33e3T^{2} \)
13 \( 1 - 31.3T + 2.19e3T^{2} \)
17 \( 1 - 34.4iT - 4.91e3T^{2} \)
19 \( 1 + 120. iT - 6.85e3T^{2} \)
23 \( 1 - 137.T + 1.21e4T^{2} \)
29 \( 1 - 93.1iT - 2.43e4T^{2} \)
31 \( 1 + 111. iT - 2.97e4T^{2} \)
37 \( 1 - 146.T + 5.06e4T^{2} \)
41 \( 1 + 8.44iT - 6.89e4T^{2} \)
43 \( 1 + 427. iT - 7.95e4T^{2} \)
47 \( 1 + 318.T + 1.03e5T^{2} \)
53 \( 1 + 291. iT - 1.48e5T^{2} \)
59 \( 1 + 364.T + 2.05e5T^{2} \)
61 \( 1 - 289.T + 2.26e5T^{2} \)
67 \( 1 + 305. iT - 3.00e5T^{2} \)
71 \( 1 + 102.T + 3.57e5T^{2} \)
73 \( 1 - 442.T + 3.89e5T^{2} \)
79 \( 1 - 245. iT - 4.93e5T^{2} \)
83 \( 1 - 478.T + 5.71e5T^{2} \)
89 \( 1 + 1.41e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.15e3T + 9.12e5T^{2} \)
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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

−8.984799102806241205640136881458, −7.955137310792556900487829183016, −7.19265601868473088551015363048, −6.78211286574200979643962746309, −5.91207131085554213880675957939, −4.82715579216067296176634939890, −3.66917118454582832268621867934, −3.02347858218736351089257276256, −2.13359299957408491201442718045, −0.52111469394317473248700534776, 0.798828659653351034334979721570, 1.67853455596717881829148258514, 2.91254707080353913066820454707, 4.13311047498426403005458387434, 5.01690051663081129155927555896, 5.50431451164367456843688190141, 6.31787086115885832260441467802, 7.79978838424907229797047668771, 8.163522119374571201073923565353, 8.983337827006868291562106741132

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