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Properties

Label 2-12e3-8.3-c2-0-54
Degree $2$
Conductor $1728$
Sign $-0.965 + 0.258i$
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  + 2.64i·5-s − 8.30i·7-s − 18.7·11-s + 22.3i·13-s + 23.4·17-s + 9.93·19-s − 32.3i·23-s + 18.0·25-s − 8.45i·29-s + 46.9i·31-s + 21.9·35-s − 28.3i·37-s − 77.7·41-s − 58.4·43-s − 54.2i·47-s + ⋯
L(s)  = 1  + 0.528i·5-s − 1.18i·7-s − 1.70·11-s + 1.71i·13-s + 1.38·17-s + 0.522·19-s − 1.40i·23-s + 0.721·25-s − 0.291i·29-s + 1.51i·31-s + 0.626·35-s − 0.765i·37-s − 1.89·41-s − 1.35·43-s − 1.15i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.965 + 0.258i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.965 + 0.258i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1765379650\)
\(L(\frac12)\) \(\approx\) \(0.1765379650\)
\(L(2)\) 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 \( 1 - 2.64iT - 25T^{2} \)
7 \( 1 + 8.30iT - 49T^{2} \)
11 \( 1 + 18.7T + 121T^{2} \)
13 \( 1 - 22.3iT - 169T^{2} \)
17 \( 1 - 23.4T + 289T^{2} \)
19 \( 1 - 9.93T + 361T^{2} \)
23 \( 1 + 32.3iT - 529T^{2} \)
29 \( 1 + 8.45iT - 841T^{2} \)
31 \( 1 - 46.9iT - 961T^{2} \)
37 \( 1 + 28.3iT - 1.36e3T^{2} \)
41 \( 1 + 77.7T + 1.68e3T^{2} \)
43 \( 1 + 58.4T + 1.84e3T^{2} \)
47 \( 1 + 54.2iT - 2.20e3T^{2} \)
53 \( 1 + 5.81iT - 2.80e3T^{2} \)
59 \( 1 + 47.5T + 3.48e3T^{2} \)
61 \( 1 + 27.7iT - 3.72e3T^{2} \)
67 \( 1 + 50.6T + 4.48e3T^{2} \)
71 \( 1 - 7.34iT - 5.04e3T^{2} \)
73 \( 1 + 29.4T + 5.32e3T^{2} \)
79 \( 1 - 43.2iT - 6.24e3T^{2} \)
83 \( 1 + 10.7T + 6.88e3T^{2} \)
89 \( 1 + 136.T + 7.92e3T^{2} \)
97 \( 1 + 54.6T + 9.40e3T^{2} \)
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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

−8.648392188072049338728929660641, −7.929197865650608386018696777481, −6.99988918747043005420871225771, −6.74852445114363047855562067586, −5.36198409543169579125321886693, −4.67896435684029530513084728542, −3.61648909586062819660570337218, −2.77080562920633822024616276381, −1.49394068675048619738666217254, −0.04670289179555109894906296833, 1.36313940661806695158444562498, 2.81775779582055457884950292593, 3.23212765904971506706841072030, 4.98395718279359079115665075998, 5.41238736434452086360645765801, 5.88253824403217645108085882242, 7.42844510440206959697459068350, 8.011957081647454874952014284402, 8.499869618034587098811088451692, 9.631220105582668923734408704303

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