Properties

Label 2-12e3-4.3-c2-0-14
Degree $2$
Conductor $1728$
Sign $-i$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.08·5-s − 6.01i·7-s + 17.7i·11-s − 12.4·13-s + 26.3·17-s + 5.19i·19-s − 29.8i·23-s − 23.8·25-s + 4.32·29-s + 44.8i·31-s − 6.49i·35-s + 20.4·37-s − 59.2·41-s + 19.1i·43-s + 41.0i·47-s + ⋯
L(s)  = 1  + 0.216·5-s − 0.859i·7-s + 1.61i·11-s − 0.955·13-s + 1.55·17-s + 0.273i·19-s − 1.29i·23-s − 0.953·25-s + 0.149·29-s + 1.44i·31-s − 0.185i·35-s + 0.551·37-s − 1.44·41-s + 0.445i·43-s + 0.873i·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.397582446\)
\(L(\frac12)\) \(\approx\) \(1.397582446\)
\(L(2)\) 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 \( 1 - 1.08T + 25T^{2} \)
7 \( 1 + 6.01iT - 49T^{2} \)
11 \( 1 - 17.7iT - 121T^{2} \)
13 \( 1 + 12.4T + 169T^{2} \)
17 \( 1 - 26.3T + 289T^{2} \)
19 \( 1 - 5.19iT - 361T^{2} \)
23 \( 1 + 29.8iT - 529T^{2} \)
29 \( 1 - 4.32T + 841T^{2} \)
31 \( 1 - 44.8iT - 961T^{2} \)
37 \( 1 - 20.4T + 1.36e3T^{2} \)
41 \( 1 + 59.2T + 1.68e3T^{2} \)
43 \( 1 - 19.1iT - 1.84e3T^{2} \)
47 \( 1 - 41.0iT - 2.20e3T^{2} \)
53 \( 1 + 70.0T + 2.80e3T^{2} \)
59 \( 1 + 28.9iT - 3.48e3T^{2} \)
61 \( 1 + 18.4T + 3.72e3T^{2} \)
67 \( 1 - 94.8iT - 4.48e3T^{2} \)
71 \( 1 - 83.8iT - 5.04e3T^{2} \)
73 \( 1 - 55.8T + 5.32e3T^{2} \)
79 \( 1 - 41.0iT - 6.24e3T^{2} \)
83 \( 1 + 35.4iT - 6.88e3T^{2} \)
89 \( 1 - 26.3T + 7.92e3T^{2} \)
97 \( 1 - 76.3T + 9.40e3T^{2} \)
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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

−9.733529362428779816534260982082, −8.441854333943751231036713036598, −7.56863952011801451401802465417, −7.13610723305663624645258704799, −6.21522278217855173531178315881, −5.04710759592555398584652642483, −4.50718567939711620544387994834, −3.43245641244953114626264210812, −2.27297692732726088702685434292, −1.18730474776151180406231443087, 0.38417128212746178585976049721, 1.82250269592757573877161536328, 2.95542134380065904348034268392, 3.66802795776475342097279068223, 5.11210966968408532300626517714, 5.67181437883752390034700338727, 6.29574703767082463216722188807, 7.60289477364322778436817585976, 8.041958798148712134837542089427, 9.049916325025588334418921513236

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