Properties

Label 2-12e3-12.11-c1-0-21
Degree $2$
Conductor $1728$
Sign $i$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.14i·5-s + 3.44i·7-s − 4.56·11-s + 6.89·13-s − 3.46i·17-s − 4.89i·19-s − 2.82·23-s − 4.89·25-s + 2.19i·29-s − 2.55i·31-s + 10.8·35-s + 4.89·37-s − 4.09i·41-s − 2.89i·43-s + 2.19·47-s + ⋯
L(s)  = 1  − 1.40i·5-s + 1.30i·7-s − 1.37·11-s + 1.91·13-s − 0.840i·17-s − 1.12i·19-s − 0.589·23-s − 0.979·25-s + 0.407i·29-s − 0.458i·31-s + 1.83·35-s + 0.805·37-s − 0.640i·41-s − 0.442i·43-s + 0.319·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.419966063\)
\(L(\frac12)\) \(\approx\) \(1.419966063\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.14iT - 5T^{2} \)
7 \( 1 - 3.44iT - 7T^{2} \)
11 \( 1 + 4.56T + 11T^{2} \)
13 \( 1 - 6.89T + 13T^{2} \)
17 \( 1 + 3.46iT - 17T^{2} \)
19 \( 1 + 4.89iT - 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 - 2.19iT - 29T^{2} \)
31 \( 1 + 2.55iT - 31T^{2} \)
37 \( 1 - 4.89T + 37T^{2} \)
41 \( 1 + 4.09iT - 41T^{2} \)
43 \( 1 + 2.89iT - 43T^{2} \)
47 \( 1 - 2.19T + 47T^{2} \)
53 \( 1 + 12.9iT - 53T^{2} \)
59 \( 1 - 2.19T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + 14.8iT - 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 7.89T + 73T^{2} \)
79 \( 1 - 2iT - 79T^{2} \)
83 \( 1 + 12.4T + 83T^{2} \)
89 \( 1 - 5.02iT - 89T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
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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

−8.989028221279246306259444890113, −8.459099993455520483534416973837, −7.88629393262715403277456088973, −6.58361082167563346850313676004, −5.53178914149489442237267755547, −5.28931870455206319304490502813, −4.26608400051538381394171792359, −2.99964976748049310157864451397, −1.97001597124107924997606638913, −0.56770975674730254520479892569, 1.33327382355754718703690081059, 2.73834587221283664102012434564, 3.64124188422156186342450232739, 4.23326907024666438766592742112, 5.81133107565770193077081844229, 6.24742469082687508108479281410, 7.20702994597106565309576660520, 7.87369788460388826029371485298, 8.449010709676954138896107970407, 9.900429735874353422952118743540

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