Properties

Label 2-12e3-8.3-c2-0-39
Degree $2$
Conductor $1728$
Sign $0.707 + 0.707i$
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  + 6i·5-s − 5i·7-s − 6·11-s + 5.19i·13-s − 31.1·17-s + 25.9·19-s − 31.1i·23-s − 11·25-s − 36i·29-s + 26i·31-s + 30·35-s − 25.9i·37-s + 20.7·43-s − 31.1i·47-s + 24·49-s + ⋯
L(s)  = 1  + 1.20i·5-s − 0.714i·7-s − 0.545·11-s + 0.399i·13-s − 1.83·17-s + 1.36·19-s − 1.35i·23-s − 0.440·25-s − 1.24i·29-s + 0.838i·31-s + 0.857·35-s − 0.702i·37-s + 0.483·43-s − 0.663i·47-s + 0.489·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.707 + 0.707i$
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: \((2,\ 1728,\ (\ :1),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.434629535\)
\(L(\frac12)\) \(\approx\) \(1.434629535\)
\(L(2)\) 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 \( 1 - 6iT - 25T^{2} \)
7 \( 1 + 5iT - 49T^{2} \)
11 \( 1 + 6T + 121T^{2} \)
13 \( 1 - 5.19iT - 169T^{2} \)
17 \( 1 + 31.1T + 289T^{2} \)
19 \( 1 - 25.9T + 361T^{2} \)
23 \( 1 + 31.1iT - 529T^{2} \)
29 \( 1 + 36iT - 841T^{2} \)
31 \( 1 - 26iT - 961T^{2} \)
37 \( 1 + 25.9iT - 1.36e3T^{2} \)
41 \( 1 + 1.68e3T^{2} \)
43 \( 1 - 20.7T + 1.84e3T^{2} \)
47 \( 1 + 31.1iT - 2.20e3T^{2} \)
53 \( 1 - 60iT - 2.80e3T^{2} \)
59 \( 1 + 18T + 3.48e3T^{2} \)
61 \( 1 + 77.9iT - 3.72e3T^{2} \)
67 \( 1 - 77.9T + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 25T + 5.32e3T^{2} \)
79 \( 1 + 31iT - 6.24e3T^{2} \)
83 \( 1 + 120T + 6.88e3T^{2} \)
89 \( 1 - 155.T + 7.92e3T^{2} \)
97 \( 1 + 85T + 9.40e3T^{2} \)
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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

−9.091066038491201475108411565217, −8.156908731168361382542524609026, −7.21883232271638412618397595704, −6.83141463896487556081011113440, −6.00053446389298124461394930974, −4.78972685707122645381240121856, −3.99925808550952381398336226064, −2.92957461216666183253174228215, −2.14702583419387544047571454795, −0.44802356264953959867370377858, 0.960563459503952240440666354666, 2.13488086233835796885912202220, 3.23455318406256749742760258440, 4.44488614821087089234793691291, 5.22623545635728580617380448273, 5.71446948665206574661276680631, 6.90607904298310569345630551630, 7.79151570429301368465591914501, 8.575926622570264907792617595693, 9.158558161106532582734850602424

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