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Properties

Label 2-12e2-144.5-c2-0-2
Degree $2$
Conductor $144$
Sign $-0.324 - 0.945i$
Analytic cond. $3.92371$
Root an. cond. $1.98083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.79 − 0.885i)2-s + (0.352 − 2.97i)3-s + (2.43 + 3.17i)4-s + (0.899 + 3.35i)5-s + (−3.26 + 5.03i)6-s + (−9.56 + 5.52i)7-s + (−1.55 − 7.84i)8-s + (−8.75 − 2.10i)9-s + (1.35 − 6.81i)10-s + (−13.6 − 3.65i)11-s + (10.3 − 6.13i)12-s + (3.24 + 12.0i)13-s + (22.0 − 1.43i)14-s + (10.3 − 1.49i)15-s + (−4.15 + 15.4i)16-s − 3.35i·17-s + ⋯
L(s)  = 1  + (−0.896 − 0.442i)2-s + (0.117 − 0.993i)3-s + (0.608 + 0.793i)4-s + (0.179 + 0.671i)5-s + (−0.544 + 0.838i)6-s + (−1.36 + 0.788i)7-s + (−0.194 − 0.980i)8-s + (−0.972 − 0.233i)9-s + (0.135 − 0.681i)10-s + (−1.24 − 0.332i)11-s + (0.859 − 0.510i)12-s + (0.249 + 0.930i)13-s + (1.57 − 0.102i)14-s + (0.688 − 0.0997i)15-s + (−0.259 + 0.965i)16-s − 0.197i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.324 - 0.945i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.324 - 0.945i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.129707 + 0.181642i\)
\(L(\frac12)\) \(\approx\) \(0.129707 + 0.181642i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.79 + 0.885i)T \)
3 \( 1 + (-0.352 + 2.97i)T \)
good5 \( 1 + (-0.899 - 3.35i)T + (-21.6 + 12.5i)T^{2} \)
7 \( 1 + (9.56 - 5.52i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (13.6 + 3.65i)T + (104. + 60.5i)T^{2} \)
13 \( 1 + (-3.24 - 12.0i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 + 3.35iT - 289T^{2} \)
19 \( 1 + (20.7 - 20.7i)T - 361iT^{2} \)
23 \( 1 + (-5.31 + 9.20i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-5.51 + 20.5i)T + (-728. - 420.5i)T^{2} \)
31 \( 1 + (7.98 - 13.8i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-5.43 - 5.43i)T + 1.36e3iT^{2} \)
41 \( 1 + (33.7 - 58.4i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-17.6 + 65.8i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-4.11 + 2.37i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (38.8 - 38.8i)T - 2.80e3iT^{2} \)
59 \( 1 + (26.1 + 97.4i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (83.6 + 22.4i)T + (3.22e3 + 1.86e3i)T^{2} \)
67 \( 1 + (-5.02 - 18.7i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 5.30T + 5.04e3T^{2} \)
73 \( 1 - 26.3iT - 5.32e3T^{2} \)
79 \( 1 + (-10.9 - 18.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (26.9 - 100. i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 - 6.78T + 7.92e3T^{2} \)
97 \( 1 + (2.33 + 4.04i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.81702256751059036503140171255, −12.27935237024485428837917540930, −11.06889328645843901532691312309, −10.09816205459430877303507523192, −8.964919309155617167346565498128, −8.049752968100556114631393887926, −6.73695466475496017505455546871, −6.14256898124200566516178850782, −3.17711292713985809465759834875, −2.25098114374426692680721192607, 0.17419560290568146585386387313, 2.97998230752497874485787923883, 4.82396813571055546047029954210, 5.98130107203064183490431159814, 7.37184437736966627759974698093, 8.613863576454357618506362334278, 9.449315903316865601599933374681, 10.38662532676648912036105736525, 10.84674020511286872059275089360, 12.74681930216248144089416950679

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