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Properties

Label 2-12e3-12.11-c3-0-46
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  + 32.9i·7-s + 89·13-s + 126. i·19-s + 125·25-s − 155. i·31-s + 433·37-s + 218. i·43-s − 740·49-s + 901·61-s − 434. i·67-s + 271·73-s − 1.31e3i·79-s + 2.92e3i·91-s − 1.85e3·97-s + 2.09e3i·103-s + ⋯
L(s)  = 1  + 1.77i·7-s + 1.89·13-s + 1.52i·19-s + 25-s − 0.903i·31-s + 1.92·37-s + 0.773i·43-s − 2.15·49-s + 1.89·61-s − 0.792i·67-s + 0.434·73-s − 1.86i·79-s + 3.37i·91-s − 1.93·97-s + 1.99i·103-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

\(L(2)\) \(\approx\) \(2.525971916\)
\(L(\frac12)\) \(\approx\) \(2.525971916\)
\(L(\frac{5}{2})\) 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 \( 1 - 125T^{2} \)
7 \( 1 - 32.9iT - 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 - 89T + 2.19e3T^{2} \)
17 \( 1 - 4.91e3T^{2} \)
19 \( 1 - 126. iT - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + 155. iT - 2.97e4T^{2} \)
37 \( 1 - 433T + 5.06e4T^{2} \)
41 \( 1 - 6.89e4T^{2} \)
43 \( 1 - 218. iT - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 901T + 2.26e5T^{2} \)
67 \( 1 + 434. iT - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 271T + 3.89e5T^{2} \)
79 \( 1 + 1.31e3iT - 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 - 7.04e5T^{2} \)
97 \( 1 + 1.85e3T + 9.12e5T^{2} \)
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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

−9.077465925801203533325430919190, −8.375593472594299980715731369817, −7.894614755145617760468743858784, −6.38051464831093955804750684542, −6.03557346717817066835255607143, −5.28874795440956191818570456493, −4.09054634318945584246150119148, −3.15883155418617383202933786699, −2.17707459238311451400203472628, −1.12290823073888404769487724260, 0.66325541633689343715607799210, 1.24292920767627478212973336230, 2.85410822194887847131114193707, 3.83747188187456173141084386753, 4.41059744504119139835553575875, 5.48711604749998189191200770512, 6.72783169248874937693317546188, 6.90158854231598245681705136473, 8.027369651572316821635617842604, 8.674445057422950011500813426253

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