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Properties

Label 1-12e2-144.133-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: $\chi_{144} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 144,\ (0:\ ),\ 0.999 - 0.0436i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.074493650 + 0.02344553485i\)
\(L(\frac12)\) \(\approx\) \(1.074493650 + 0.02344553485i\)
\(L(1)\) \(\approx\) \(1.041603235 + 0.01562110410i\)
\(L(1)\) \(\approx\) \(1.041603235 + 0.01562110410i\)

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

−27.739645609855251116560848757420, −27.580338787743097623928579434849, −26.37950136461734151038209436271, −25.42090043962155236708155828537, −24.230023457713369950309195704364, −23.03039873979269296530933735582, −22.83714711386470640072483797920, −21.16366834985297456215270159446, −20.277094670580805074900409265901, −19.33694547753677945919958074726, −18.30415435204583581336176415208, −17.172241720962278228928954782155, −16.172550427556509337148336030666, −14.91013364619145371713133541740, −14.243839754696670642596571240878, −12.84944854836732289811520202801, −11.59987852014174934875586974710, −10.83787930335772744616862674077, −9.5968369987962211174245654440, −7.982898776254070079850738472385, −7.36056677918190970502294192483, −5.92498122455540931618521070404, −4.22229936526961315370068268315, −3.43895733425069919901904275690, −1.344176691939833784628223081592, 1.33928834986559961061617885769, 3.23111239798975939489190497077, 4.4965156022251057808377553462, 5.73205128707948637549220863770, 7.15816726989768012691740489392, 8.57167141245576255553884520918, 9.03718229034029494376168077576, 10.963580318504089200896032865819, 11.753299840628310334450454557376, 12.66543757883296830174171833776, 14.10794597270990985921940919850, 15.1226886296748420560948100621, 16.12804695144082352252450562517, 17.01896612525299316872624951108, 18.473071540390553572104270030562, 19.160206270027812507404216580, 20.32687485442670263679580815500, 21.27809537730038635596839525475, 22.29599873721056930229501277749, 23.50384025547221616396882704483, 24.27732922845888841114055264105, 25.16728554077660006599325731430, 26.36364950114046252147768566271, 27.66225742735853770699571694072, 27.90375607730545458735367025750

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