↑

Properties

Label 2-12e3-9.5-c2-0-25
Degree $2$
Conductor $1728$
Sign $0.939 + 0.342i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (3 + 1.73i)5-s + (−1 − 1.73i)7-s + (−1.5 + 0.866i)11-s + (−2 + 3.46i)13-s − 15.5i·17-s − 11·19-s + (24 + 13.8i)23-s + (−6.5 − 11.2i)25-s + (39 − 22.5i)29-s + (−16 + 27.7i)31-s − 6.92i·35-s + 34·37-s + (10.5 + 6.06i)41-s + (−30.5 − 52.8i)43-s + (42 − 24.2i)47-s + ⋯
L(s)  = 1  + (0.600 + 0.346i)5-s + (−0.142 − 0.247i)7-s + (−0.136 + 0.0787i)11-s + (−0.153 + 0.266i)13-s − 0.916i·17-s − 0.578·19-s + (1.04 + 0.602i)23-s + (−0.260 − 0.450i)25-s + (1.34 − 0.776i)29-s + (−0.516 + 0.893i)31-s − 0.197i·35-s + 0.918·37-s + (0.256 + 0.147i)41-s + (−0.709 − 1.22i)43-s + (0.893 − 0.515i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.132040071\)
\(L(\frac12)\) \(\approx\) \(2.132040071\)
\(L(2)\) 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 \( 1 + (-3 - 1.73i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (2 - 3.46i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 15.5iT - 289T^{2} \)
19 \( 1 + 11T + 361T^{2} \)
23 \( 1 + (-24 - 13.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-39 + 22.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (16 - 27.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 34T + 1.36e3T^{2} \)
41 \( 1 + (-10.5 - 6.06i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (30.5 + 52.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-42 + 24.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (-43.5 - 25.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-28 - 48.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (15.5 - 26.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 31.1iT - 5.04e3T^{2} \)
73 \( 1 - 65T + 5.32e3T^{2} \)
79 \( 1 + (19 + 32.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (42 - 24.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 124. iT - 7.92e3T^{2} \)
97 \( 1 + (-57.5 - 99.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.103360344207159898920610425777, −8.422281559293608313615749913780, −7.27677162364663407689248706964, −6.82258628259821550718471267515, −5.87903217090406854249127397409, −5.03583579956130266918293455272, −4.10352989142998244908413672799, −2.94626641636934781095402489316, −2.12577039682167152336510740947, −0.70781884081582184436872908377, 0.931844805339712527937796309842, 2.12870615818246482858305601230, 3.09488518747451333550429056011, 4.28079031030023182528979085967, 5.13934782307314694319110860547, 5.98211061803480452218295339703, 6.61978533163848937511124671488, 7.70208235131836219831277131211, 8.475838666262513109969994039596, 9.175452967622026513291174850542

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