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Properties

Label 1-12e2-144.5-r1-0-0
Degree $1$
Conductor $144$
Sign $-0.675 - 0.737i$
Analytic cond. $15.4749$
Root an. cond. $15.4749$
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

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

Invariants

Degree: \(1\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.675 - 0.737i$
Analytic conductor: \(15.4749\)
Root analytic conductor: \(15.4749\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 144,\ (1:\ ),\ -0.675 - 0.737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2439322442 - 0.5543781958i\)
\(L(\frac12)\) \(\approx\) \(0.2439322442 - 0.5543781958i\)
\(L(1)\) \(\approx\) \(0.7819640992 - 0.1015051505i\)
\(L(1)\) \(\approx\) \(0.7819640992 - 0.1015051505i\)

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

−28.17805570193971825564518339788, −27.42050558858268121072741258690, −26.75309962059714336138358105368, −25.073005646355769878882497536197, −24.573042756620214908322160483901, −23.58912986063708882075794091050, −22.34395221533004562415473652871, −21.57488952144112309586504122763, −20.223494570918140527521777108723, −19.53751115728851469229202676070, −18.42348096549332580054723930659, −17.277517646611619198234352101200, −16.19508036650998977449987109799, −15.20757563730563165864495920151, −14.32773804421881269108430743534, −12.75273916807179084536291445521, −11.928321870391466713762515480353, −11.06010538997331362979227688725, −9.377926516788338933969920185036, −8.46793032056645228627705909805, −7.43353508212745205849562539604, −5.84796658048811611194972251181, −4.68381656041429843550995439817, −3.38916011253373315173678801756, −1.64530248691248477654477491440, 0.22945317080313966746797483951, 2.1592381859616388000489147715, 3.906854071514673899994380973068, 4.66118526224735817026487024387, 6.732800511659966729336733816441, 7.34081424584046495718115687237, 8.67903775426812063679156194891, 10.050027362176073889005183853728, 11.22771194299568590302193681427, 11.93691999962699445908810323923, 13.41595478087696420064413068291, 14.57107373116436504818864766247, 15.2628225694481446661130474004, 16.67436638881811098221230663238, 17.49069056912276681821930552028, 18.72658655780692126598763951724, 19.844630273113050719228519706352, 20.34067954975110090989785889888, 22.00334411447941135407049130784, 22.60876253864297753606687914285, 23.8862356576150577589458632580, 24.37300288633915568806110382606, 25.978473922161208799421792842303, 26.74379113610358127356011650740, 27.476013147212684071020504130738

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