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Properties

Label 1-12e2-144.61-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: $\chi_{144} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 144,\ (0:\ ),\ 0.737 + 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.150250945 + 0.4473081288i\)
\(L(\frac12)\) \(\approx\) \(1.150250945 + 0.4473081288i\)
\(L(1)\) \(\approx\) \(1.150827809 + 0.2347359201i\)
\(L(1)\) \(\approx\) \(1.150827809 + 0.2347359201i\)

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

−28.22409976261780974937461297939, −27.10812850803127864715966644477, −26.19317783574479052109944253696, −25.23850884003800595368644315089, −24.03157371742785548696589124722, −23.571310597998987771132760806785, −21.94838347745724414320963566059, −21.21084414098925093113362060555, −20.353231108610435032182567712725, −19.19400096602144880625450695694, −17.91574908313013314797945267240, −17.08656534033987571083453096671, −16.27243855607965298458769032026, −14.76327078909592812211936599294, −13.74529635679100436743086367477, −13.010651436283290282815048477741, −11.5931558534562405543571810469, −10.39329926275377908402457717490, −9.47742326869040017364022950254, −8.14218468185992726511231678128, −7.01058019415968811346358300917, −5.49316990945273294696190586895, −4.60536507056324465500897908646, −2.816647569870137953116536537406, −1.24858284831821349740688270975, 1.94978390999905226049172420079, 2.965364352181745798535297775473, 5.01085381945321022388314072010, 5.79210664772512344621176632938, 7.25745569529254896926545042143, 8.42891351521533535034748678482, 9.83686005473442091671020540783, 10.53440740789633100013840530694, 12.06046077331943052070076842248, 12.912141330384159890119348036735, 14.40162559781193508745857225000, 14.89573581236339535401328132814, 16.27991990278613282341750887264, 17.58680646703326072975754082507, 18.23019981904451896514554033878, 19.187971632564422626229963053429, 20.7688218715359179138966423564, 21.334874614737078409226028830726, 22.399057083405768067425982595468, 23.33335879495198781870197747724, 24.82944219151530953996661367079, 25.21286558916274754325313450055, 26.39571370907086174707160166842, 27.38584030137224282484194120859, 28.50906130374281762528817284875

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