Properties

Degree $2$
Conductor $144$
Sign $0.987 - 0.160i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.89 + 0.795i)3-s + (0.355 − 0.615i)5-s + (2.70 − 1.56i)7-s + (7.73 + 4.60i)9-s + (14.3 − 8.30i)11-s + (−9.17 + 15.8i)13-s + (1.51 − 1.49i)15-s − 9.69·17-s − 8.20i·19-s + (9.06 − 2.36i)21-s + (1.94 + 1.12i)23-s + (12.2 + 21.2i)25-s + (18.7 + 19.4i)27-s + (−20.8 − 36.0i)29-s + (−21.6 − 12.4i)31-s + ⋯
L(s)  = 1  + (0.964 + 0.265i)3-s + (0.0710 − 0.123i)5-s + (0.386 − 0.223i)7-s + (0.859 + 0.511i)9-s + (1.30 − 0.754i)11-s + (−0.705 + 1.22i)13-s + (0.101 − 0.0998i)15-s − 0.570·17-s − 0.431i·19-s + (0.431 − 0.112i)21-s + (0.0847 + 0.0489i)23-s + (0.489 + 0.848i)25-s + (0.692 + 0.721i)27-s + (−0.717 − 1.24i)29-s + (−0.697 − 0.402i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.987 - 0.160i$
Motivic weight: \(2\)
Character: $\chi_{144} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.987 - 0.160i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.03382 + 0.164129i\)
\(L(\frac12)\) \(\approx\) \(2.03382 + 0.164129i\)
\(L(2)\) 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 \( 1 + (-2.89 - 0.795i)T \)
good5 \( 1 + (-0.355 + 0.615i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-2.70 + 1.56i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-14.3 + 8.30i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (9.17 - 15.8i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 9.69T + 289T^{2} \)
19 \( 1 + 8.20iT - 361T^{2} \)
23 \( 1 + (-1.94 - 1.12i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (20.8 + 36.0i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (21.6 + 12.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 40.3T + 1.36e3T^{2} \)
41 \( 1 + (-25.6 + 44.5i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (56.6 - 32.7i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-29.2 + 16.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 90.6T + 2.80e3T^{2} \)
59 \( 1 + (66.2 + 38.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-1.35 - 2.35i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (34.5 + 19.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 102. iT - 5.04e3T^{2} \)
73 \( 1 - 38.1T + 5.32e3T^{2} \)
79 \( 1 + (-94.4 + 54.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-113. + 65.5i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 38.0T + 7.92e3T^{2} \)
97 \( 1 + (12.1 + 21.1i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.14518606936210922730987922780, −11.78929236750350372327190914361, −10.90084714013575732912650348198, −9.358124294585088064590385643796, −9.034371580288777622773984902127, −7.66859457676529520858103927629, −6.57910773655247702585934258984, −4.73323529336662416224810390004, −3.64502217604008860925036192930, −1.86023432328296560219928517088, 1.78725362993032460732753634958, 3.34920815127489152435922556287, 4.82483319495542460263399861402, 6.58558407819538266621258593217, 7.57907542402849084219650136688, 8.682089879794007515344825621258, 9.574838340118705500773704143947, 10.66782845464260303652360659113, 12.16997789574151613310148816273, 12.72871405078269697930313691711

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