Properties

Label 2-12e3-12.11-c1-0-29
Degree $2$
Conductor $1728$
Sign $-1$
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  − 3i·5-s − 1.73i·7-s − 5.19·11-s + 2·13-s + 6i·17-s − 6.92i·19-s − 4·25-s − 6i·29-s − 5.19i·31-s − 5.19·35-s − 8·37-s + 10.3i·43-s − 10.3·47-s + 4·49-s + 9i·53-s + ⋯
L(s)  = 1  − 1.34i·5-s − 0.654i·7-s − 1.56·11-s + 0.554·13-s + 1.45i·17-s − 1.58i·19-s − 0.800·25-s − 1.11i·29-s − 0.933i·31-s − 0.878·35-s − 1.31·37-s + 1.58i·43-s − 1.51·47-s + 0.571·49-s + 1.23i·53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-1$
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: \((2,\ 1728,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7298842198\)
\(L(\frac12)\) \(\approx\) \(0.7298842198\)
\(L(\frac{3}{2})\) 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 \( 1 + 3iT - 5T^{2} \)
7 \( 1 + 1.73iT - 7T^{2} \)
11 \( 1 + 5.19T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 6.92iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + 5.19iT - 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + 3.46iT - 79T^{2} \)
83 \( 1 - 5.19T + 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
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

−8.826024807460950607352671764792, −8.143311529025134716401097355401, −7.61407736051520757172702792508, −6.42661248408913616280020231126, −5.55924040153964734078941035547, −4.73923558901712691720188555035, −4.11136131290627587473294741885, −2.81466538046308575267979916916, −1.50371005452550506370348559514, −0.26480066013579019922542109874, 1.93532531078738022073602960429, 2.95049089177600711613217247057, 3.47918778376496003955778359435, 5.07212918553536742522905934743, 5.58850252708836668728060585045, 6.64471895768684495816932025495, 7.26718451181649366410742070018, 8.089367699753014909268375469234, 8.850656792785135421373551271300, 9.930027860114864397234528088027

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