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Properties

Label 2-12e2-16.13-c1-0-2
Degree $2$
Conductor $144$
Sign $0.946 + 0.324i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.38 + 0.297i)2-s + (1.82 − 0.822i)4-s + (0.595 − 0.595i)5-s − 1.64i·7-s + (−2.27 + 1.68i)8-s + (−0.645 + i)10-s + (3.36 − 3.36i)11-s + (2.64 + 2.64i)13-s + (0.489 + 2.27i)14-s + (2.64 − 3i)16-s + 5.53·17-s + (−3.64 − 3.64i)19-s + (0.595 − 1.57i)20-s + (−3.64 + 5.64i)22-s + 4.33i·23-s + ⋯
L(s)  = 1  + (−0.977 + 0.210i)2-s + (0.911 − 0.411i)4-s + (0.266 − 0.266i)5-s − 0.622i·7-s + (−0.804 + 0.594i)8-s + (−0.204 + 0.316i)10-s + (1.01 − 1.01i)11-s + (0.733 + 0.733i)13-s + (0.130 + 0.608i)14-s + (0.661 − 0.750i)16-s + 1.34·17-s + (−0.836 − 0.836i)19-s + (0.133 − 0.352i)20-s + (−0.777 + 1.20i)22-s + 0.904i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.946 + 0.324i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.946 + 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.795340 - 0.132449i\)
\(L(\frac12)\) \(\approx\) \(0.795340 - 0.132449i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.297i)T \)
3 \( 1 \)
good5 \( 1 + (-0.595 + 0.595i)T - 5iT^{2} \)
7 \( 1 + 1.64iT - 7T^{2} \)
11 \( 1 + (-3.36 + 3.36i)T - 11iT^{2} \)
13 \( 1 + (-2.64 - 2.64i)T + 13iT^{2} \)
17 \( 1 - 5.53T + 17T^{2} \)
19 \( 1 + (3.64 + 3.64i)T + 19iT^{2} \)
23 \( 1 - 4.33iT - 23T^{2} \)
29 \( 1 + (6.12 + 6.12i)T + 29iT^{2} \)
31 \( 1 + 5.64T + 31T^{2} \)
37 \( 1 + (0.645 - 0.645i)T - 37iT^{2} \)
41 \( 1 - 7.91iT - 41T^{2} \)
43 \( 1 + (0.354 - 0.354i)T - 43iT^{2} \)
47 \( 1 + 9.10T + 47T^{2} \)
53 \( 1 + (4.93 - 4.93i)T - 53iT^{2} \)
59 \( 1 + (-4.33 + 4.33i)T - 59iT^{2} \)
61 \( 1 + (0.645 + 0.645i)T + 61iT^{2} \)
67 \( 1 + (-4 - 4i)T + 67iT^{2} \)
71 \( 1 - 13.4iT - 71T^{2} \)
73 \( 1 + 3.29iT - 73T^{2} \)
79 \( 1 - 9.64T + 79T^{2} \)
83 \( 1 + (3.36 + 3.36i)T + 83iT^{2} \)
89 \( 1 - 2.38iT - 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−13.13600465982306969696716284655, −11.55154726853277359036285225184, −11.08832704975685869313095755658, −9.702436023312699989000049770099, −9.012882607003931322219966642917, −7.88624222948423358551812072092, −6.70165286201588683554538088979, −5.67822374745884869217378005178, −3.63882502185942196668510611947, −1.36730300352931106347058633579, 1.86482816044526920355205083112, 3.58150219137325142690961038814, 5.76594469927182401408491685621, 6.84872811845345983336072193087, 8.088744057848689508601192153748, 9.071795243332461926822170683912, 10.06130592737206774047143225297, 10.87888124592579731942243653963, 12.20777652045142093530517887449, 12.62853094342273135041684188470

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