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Properties

Label 1-12e2-144.83-r0-0-0
Degree $1$
Conductor $144$
Sign $-0.737 - 0.675i$
Analytic cond. $0.668733$
Root an. cond. $0.668733$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)5-s + (−0.5 − 0.866i)7-s + (−0.866 + 0.5i)11-s + (−0.866 − 0.5i)13-s − 17-s − i·19-s + (0.5 − 0.866i)23-s + (0.5 + 0.866i)25-s + (−0.866 + 0.5i)29-s + (0.5 − 0.866i)31-s − i·35-s − i·37-s + (−0.5 + 0.866i)41-s + (0.866 − 0.5i)43-s + (−0.5 − 0.866i)47-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)5-s + (−0.5 − 0.866i)7-s + (−0.866 + 0.5i)11-s + (−0.866 − 0.5i)13-s − 17-s − i·19-s + (0.5 − 0.866i)23-s + (0.5 + 0.866i)25-s + (−0.866 + 0.5i)29-s + (0.5 − 0.866i)31-s − i·35-s − i·37-s + (−0.5 + 0.866i)41-s + (0.866 − 0.5i)43-s + (−0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.737 - 0.675i$
Analytic conductor: \(0.668733\)
Root analytic conductor: \(0.668733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 144,\ (0:\ ),\ -0.737 - 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1680405356 - 0.4321155206i\)
\(L(\frac12)\) \(\approx\) \(0.1680405356 - 0.4321155206i\)
\(L(1)\) \(\approx\) \(0.6277966526 - 0.2247332667i\)
\(L(1)\) \(\approx\) \(0.6277966526 - 0.2247332667i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 - T \)
19 \( 1 - iT \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.79680594511973345162444902468, −27.51969071755036383876417884976, −26.71889368006214343150898759249, −25.84132355613440015179892960906, −24.61660133012802569322232391107, −23.73357711675888642995527322279, −22.63223192161968416378055887547, −21.88442323167343940931146188135, −20.72730550710202482680194532237, −19.26369504791629783379859768945, −18.97610493201681201099636845587, −17.73302819166405191591859344110, −16.266343958691382526210814969882, −15.52738827983028136158685250107, −14.61104744976331119356051814822, −13.213337381011280110465292085532, −12.10058557210057877745417150494, −11.21994198323372948880512710105, −9.9594532901678969070846992240, −8.683472798736389899164746822131, −7.576479168840617511376121056457, −6.39054266666564286342436988766, −5.038012186945144481609120004725, −3.523221594778210757124065519001, −2.3852392823855482691244633398, 0.382664198810953685768863750, 2.630399263613704167981861257, 4.132097550971125567453353721983, 5.068291514043344752924122332622, 6.92274408392682492236120765455, 7.69700441122847807241879801899, 9.01042641756596972529775171696, 10.28116395273206158819564778900, 11.29635434718694868762847362976, 12.69736265436530494007126295434, 13.240070605725596333929547735876, 14.88203472786541390326528722074, 15.72715614658761812643568927335, 16.74028802269945606267574156813, 17.72089111141267059900524212706, 19.097839003430207473691431016519, 20.03730351875213638761950585438, 20.56671005921762058789589189944, 22.1480662637107465730307773026, 23.0395838547736654758279026233, 23.90011111683052007752916314610, 24.763355979028446593167483690894, 26.26147754741166001911669460712, 26.72286626323540396257359795181, 27.96831766333819529726629881518

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