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Properties

Label 2-12e3-12.11-c3-0-78
Degree $2$
Conductor $1728$
Sign $i$
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  + 13.1i·5-s − 4.49i·7-s + 22.3·11-s + 73.5·13-s − 42.6i·17-s − 122. i·19-s − 197.·23-s − 49.2·25-s − 14.6i·29-s − 147. i·31-s + 59.2·35-s − 234.·37-s − 396. i·41-s + 280. i·43-s − 534.·47-s + ⋯
L(s)  = 1  + 1.18i·5-s − 0.242i·7-s + 0.612·11-s + 1.56·13-s − 0.608i·17-s − 1.47i·19-s − 1.79·23-s − 0.393·25-s − 0.0940i·29-s − 0.852i·31-s + 0.286·35-s − 1.04·37-s − 1.50i·41-s + 0.995i·43-s − 1.65·47-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(2)\) \(\approx\) \(1.350092215\)
\(L(\frac12)\) \(\approx\) \(1.350092215\)
\(L(\frac{5}{2})\) 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 \( 1 - 13.1iT - 125T^{2} \)
7 \( 1 + 4.49iT - 343T^{2} \)
11 \( 1 - 22.3T + 1.33e3T^{2} \)
13 \( 1 - 73.5T + 2.19e3T^{2} \)
17 \( 1 + 42.6iT - 4.91e3T^{2} \)
19 \( 1 + 122. iT - 6.85e3T^{2} \)
23 \( 1 + 197.T + 1.21e4T^{2} \)
29 \( 1 + 14.6iT - 2.43e4T^{2} \)
31 \( 1 + 147. iT - 2.97e4T^{2} \)
37 \( 1 + 234.T + 5.06e4T^{2} \)
41 \( 1 + 396. iT - 6.89e4T^{2} \)
43 \( 1 - 280. iT - 7.95e4T^{2} \)
47 \( 1 + 534.T + 1.03e5T^{2} \)
53 \( 1 - 337. iT - 1.48e5T^{2} \)
59 \( 1 + 672.T + 2.05e5T^{2} \)
61 \( 1 - 80.8T + 2.26e5T^{2} \)
67 \( 1 + 251. iT - 3.00e5T^{2} \)
71 \( 1 - 95.8T + 3.57e5T^{2} \)
73 \( 1 + 251.T + 3.89e5T^{2} \)
79 \( 1 + 499. iT - 4.93e5T^{2} \)
83 \( 1 - 16.1T + 5.71e5T^{2} \)
89 \( 1 - 321. iT - 7.04e5T^{2} \)
97 \( 1 + 210.T + 9.12e5T^{2} \)
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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

−8.808413876483025101502312202841, −7.899678436633747533054962949549, −7.03693937804291275802290519977, −6.44868557110806247294483995216, −5.75586771949499343288092434795, −4.43192539348139677762515589524, −3.63776739890070005855471255753, −2.80426522561841810367831385698, −1.66158833478226911071082036854, −0.28915285894052177901049157177, 1.23517694999594514150464862171, 1.75778430987209610408677528963, 3.53018457049056108514049935628, 4.04231608452586400799286671786, 5.12694863637537294004348892680, 5.97260620103225914181919270974, 6.50091905325774426582816669630, 7.933710827851882393855958039504, 8.419956023782091760359641406547, 8.939962023384485714724465134465

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