Properties

Degree $2$
Conductor $144$
Sign $0.961 + 0.274i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.64 − 1.40i)3-s + (−3.01 + 5.22i)5-s + (10.2 − 5.90i)7-s + (5.04 + 7.45i)9-s + (5.28 − 3.05i)11-s + (7.44 − 12.9i)13-s + (15.3 − 9.60i)15-s + 26.6·17-s + 9.45i·19-s + (−35.4 + 1.25i)21-s + (−17.2 − 9.96i)23-s + (−5.70 − 9.88i)25-s + (−2.86 − 26.8i)27-s + (22.3 + 38.6i)29-s + (5.42 + 3.13i)31-s + ⋯
L(s)  = 1  + (−0.883 − 0.469i)3-s + (−0.603 + 1.04i)5-s + (1.46 − 0.844i)7-s + (0.560 + 0.828i)9-s + (0.480 − 0.277i)11-s + (0.572 − 0.992i)13-s + (1.02 − 0.640i)15-s + 1.57·17-s + 0.497i·19-s + (−1.68 + 0.0597i)21-s + (−0.750 − 0.433i)23-s + (−0.228 − 0.395i)25-s + (−0.106 − 0.994i)27-s + (0.769 + 1.33i)29-s + (0.174 + 0.101i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.961 + 0.274i$
Motivic weight: \(2\)
Character: $\chi_{144} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.961 + 0.274i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.18924 - 0.166520i\)
\(L(\frac12)\) \(\approx\) \(1.18924 - 0.166520i\)
\(L(2)\) 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 \( 1 + (2.64 + 1.40i)T \)
good5 \( 1 + (3.01 - 5.22i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-10.2 + 5.90i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-5.28 + 3.05i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-7.44 + 12.9i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 26.6T + 289T^{2} \)
19 \( 1 - 9.45iT - 361T^{2} \)
23 \( 1 + (17.2 + 9.96i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-22.3 - 38.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-5.42 - 3.13i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 6.65T + 1.36e3T^{2} \)
41 \( 1 + (-8.82 + 15.2i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-20.2 + 11.7i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (36.4 - 21.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 51.6T + 2.80e3T^{2} \)
59 \( 1 + (32.9 + 18.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (45.3 + 78.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-53.4 - 30.8i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 39.5iT - 5.04e3T^{2} \)
73 \( 1 - 35.0T + 5.32e3T^{2} \)
79 \( 1 + (77.9 - 45.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-102. + 59.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 14.4T + 7.92e3T^{2} \)
97 \( 1 + (-67.5 - 117. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.55962627535398259700202884561, −11.66340254825863636577302124687, −10.80963940141530266551179191178, −10.37369682082433646958955655313, −8.063533763925235125157699197932, −7.59191992224144548399054985124, −6.37096570977863695465809638969, −5.07253490157101743692584781237, −3.57716111407378882740140136180, −1.20968348956690046311347690650, 1.33410317274169736428023879096, 4.18049605068638595755532732686, 4.95579028941721166669064100603, 6.06105153963517361773823616850, 7.79624454824661610368919736938, 8.747353500773178456461133305740, 9.775288689755512118886476469442, 11.29535166996401693791930532170, 11.84034443494861347806719901667, 12.37710353599033587310446254629

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