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Properties

Label 1-12e2-144.11-r0-0-0
Degree $1$
Conductor $144$
Sign $0.999 - 0.0436i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.999 - 0.0436i$
Analytic conductor: \(0.668733\)
Root analytic conductor: \(0.668733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 144,\ (0:\ ),\ 0.999 - 0.0436i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.246029576 + 0.02718846207i\)
\(L(\frac12)\) \(\approx\) \(1.246029576 + 0.02718846207i\)
\(L(1)\) \(\approx\) \(1.180004663 + 0.008500080919i\)
\(L(1)\) \(\approx\) \(1.180004663 + 0.008500080919i\)

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

−28.38036811172214515165290703943, −27.52637491732051560960299321772, −26.04329151574493825172218502272, −25.23621948958685598989862937368, −24.6680768061525680804606707954, −23.2910903678040975263650871663, −22.13626011724790580505295867431, −21.50716039371791485203612736271, −20.28316055542247093892355418198, −19.3983981116233155861711999006, −17.98331138819604457175672293138, −17.43195514657326531520617152961, −16.05406403341586120178155275517, −15.225256803304280130463146125688, −13.80733681485617309082164563154, −12.95582308108077315138375222996, −11.94089847657189504146064016554, −10.56869618378654159570826378255, −9.20863731853483441255131571435, −8.78077549554880910313002012639, −6.83961927401034900876028184052, −5.88569533485788445622561783972, −4.69111115441619014578502061324, −2.97973117602483753210825156700, −1.55860514360327138567531184302, 1.47299113319574495542064452947, 3.13664539003776701449880614918, 4.37646893479330905747355060689, 6.26222322533545082156185205182, 6.65350288161047669491358035185, 8.40316159750593448265246327090, 9.617429093401369873562438457007, 10.55328878824232771915053580216, 11.60854100696172865342990493643, 13.25825849470781638953761226248, 13.80555568011575770537164463591, 14.889061586461488086333427067250, 16.399499264058321846687809833395, 17.068402303736141345293297828385, 18.26536003226120732424939676969, 19.197211622054919809905390141067, 20.35689099588115701680747968783, 21.3468535984470721472665224158, 22.39965811137935353998119292059, 23.13007976961770827690018253213, 24.46591566219346209587749836927, 25.357417606965001761363501520587, 26.3807997623127450759895485668, 26.991487148840363393918798630129, 28.4930717383323597745555366868

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