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Properties

Label 2-12e2-16.13-c1-0-6
Degree $2$
Conductor $144$
Sign $-0.0819 + 0.996i$
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  + (−0.874 − 1.11i)2-s + (−0.470 + 1.94i)4-s + (0.334 − 0.334i)5-s − 4.55i·7-s + (2.57 − 1.17i)8-s + (−0.665 − 0.0793i)10-s + (2.47 − 2.47i)11-s + (−0.0594 − 0.0594i)13-s + (−5.06 + 3.98i)14-s + (−3.55 − 1.82i)16-s − 3.61·17-s + (2.55 + 2.55i)19-s + (0.493 + 0.808i)20-s + (−4.91 − 0.585i)22-s − 2.82i·23-s + ⋯
L(s)  = 1  + (−0.618 − 0.785i)2-s + (−0.235 + 0.971i)4-s + (0.149 − 0.149i)5-s − 1.72i·7-s + (0.909 − 0.416i)8-s + (−0.210 − 0.0250i)10-s + (0.745 − 0.745i)11-s + (−0.0164 − 0.0164i)13-s + (−1.35 + 1.06i)14-s + (−0.889 − 0.457i)16-s − 0.877·17-s + (0.586 + 0.586i)19-s + (0.110 + 0.180i)20-s + (−1.04 − 0.124i)22-s − 0.589i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0819 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0819 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.0819 + 0.996i$
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: \((2,\ 144,\ (\ :1/2),\ -0.0819 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.557661 - 0.605397i\)
\(L(\frac12)\) \(\approx\) \(0.557661 - 0.605397i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.874 + 1.11i)T \)
3 \( 1 \)
good5 \( 1 + (-0.334 + 0.334i)T - 5iT^{2} \)
7 \( 1 + 4.55iT - 7T^{2} \)
11 \( 1 + (-2.47 + 2.47i)T - 11iT^{2} \)
13 \( 1 + (0.0594 + 0.0594i)T + 13iT^{2} \)
17 \( 1 + 3.61T + 17T^{2} \)
19 \( 1 + (-2.55 - 2.55i)T + 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (-5.16 - 5.16i)T + 29iT^{2} \)
31 \( 1 + 0.557T + 31T^{2} \)
37 \( 1 + (-4.38 + 4.38i)T - 37iT^{2} \)
41 \( 1 - 9.27iT - 41T^{2} \)
43 \( 1 + (1.61 - 1.61i)T - 43iT^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + (-0.493 + 0.493i)T - 53iT^{2} \)
59 \( 1 + (4 - 4i)T - 59iT^{2} \)
61 \( 1 + (-2.72 - 2.72i)T + 61iT^{2} \)
67 \( 1 + (-3.77 - 3.77i)T + 67iT^{2} \)
71 \( 1 + 9.11iT - 71T^{2} \)
73 \( 1 + 0.541iT - 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + (-10.6 - 10.6i)T + 83iT^{2} \)
89 \( 1 + 14.6iT - 89T^{2} \)
97 \( 1 - 4.31T + 97T^{2} \)
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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

−12.85162254339632077805363363778, −11.51990891354610645510553951436, −10.80575187557676513041812197989, −9.898794347111091196359100260108, −8.837513673972729298388017233276, −7.68945324302865739015237342679, −6.63853057305733401861989582337, −4.49491056660809135317068564670, −3.39032603870833149652686017146, −1.15700293506763107024784850141, 2.23527179801766438322604082817, 4.74405947475563852257880690753, 5.95194365995192316877196634557, 6.85995740379068678080858340147, 8.283307167583170434190374139073, 9.148276151251859392829740059007, 9.881547212685199822550377995882, 11.35884616663681694960786669916, 12.23160945701819796624621358366, 13.59697298977724916445531051173

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