Properties

Degree $2$
Conductor $144$
Sign $0.342 - 0.939i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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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

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.342 - 0.939i$
Motivic weight: \(1\)
Character: $\chi_{144} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.941878 + 0.659510i\)
\(L(\frac12)\) \(\approx\) \(0.941878 + 0.659510i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

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

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