Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} $
Sign $0.342 + 0.939i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·3-s + (1.5 − 0.866i)5-s + (−1.5 − 0.866i)7-s − 2.99·9-s + (1.5 − 2.59i)11-s + (2.5 + 4.33i)13-s + (−1.49 − 2.59i)15-s − 6.92i·17-s + 3.46i·19-s + (−1.49 + 2.59i)21-s + (4.5 + 7.79i)23-s + (−1 + 1.73i)25-s + 5.19i·27-s + (1.5 + 0.866i)29-s + (−4.5 + 2.59i)31-s + ⋯
L(s)  = 1  − 0.999i·3-s + (0.670 − 0.387i)5-s + (−0.566 − 0.327i)7-s − 0.999·9-s + (0.452 − 0.783i)11-s + (0.693 + 1.20i)13-s + (−0.387 − 0.670i)15-s − 1.68i·17-s + 0.794i·19-s + (−0.327 + 0.566i)21-s + (0.938 + 1.62i)23-s + (−0.200 + 0.346i)25-s + 0.999i·27-s + (0.278 + 0.160i)29-s + (−0.808 + 0.466i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(144\)    =    \(2^{4} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.342 + 0.939i$
motivic weight  =  \(1\)
character  :  $\chi_{144} (47, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 144,\ (\ :1/2),\ 0.342 + 0.939i)\)
\(L(1)\)  \(\approx\)  \(0.941878 - 0.659510i\)
\(L(\frac12)\)  \(\approx\)  \(0.941878 - 0.659510i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 1.73iT \)
good5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.5 + 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.92iT - 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + (-4.5 - 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 0.866i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.5 - 2.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.5 - 2.59i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.5 + 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (7.5 + 4.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.5 + 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.22180926602360114803102296989, −11.87905048699547129785020985695, −11.20291703544438106782868112281, −9.511677227347317335941811743246, −8.885870633594408685217822171037, −7.39513206385297268528046215994, −6.46462364300244324935226542980, −5.39849902506889789807652657025, −3.35383496393844425103258497966, −1.43834980079011542247287906633, 2.71520314460127089964457787895, 4.14480418184802737496489738642, 5.65049118824998910766678737467, 6.53807913591060777441140676272, 8.361474323775439297448556447842, 9.312944172572840639402570621401, 10.34028644547091426521377671830, 10.84736513457373417686869429970, 12.40152109552675379619288166045, 13.24115208490608309117435303920

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