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Properties

Label 1-111-111.32-r0-0-0
Degree $1$
Conductor $111$
Sign $-0.994 + 0.103i$
Analytic cond. $0.515481$
Root an. cond. $0.515481$
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  + (−0.642 − 0.766i)2-s + (−0.173 + 0.984i)4-s + (−0.342 − 0.939i)5-s + (−0.939 + 0.342i)7-s + (0.866 − 0.5i)8-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.984 − 0.173i)13-s + (0.866 + 0.5i)14-s + (−0.939 − 0.342i)16-s + (−0.984 + 0.173i)17-s + (0.642 − 0.766i)19-s + (0.984 − 0.173i)20-s + (−0.342 + 0.939i)22-s + (−0.866 − 0.5i)23-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)2-s + (−0.173 + 0.984i)4-s + (−0.342 − 0.939i)5-s + (−0.939 + 0.342i)7-s + (0.866 − 0.5i)8-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.984 − 0.173i)13-s + (0.866 + 0.5i)14-s + (−0.939 − 0.342i)16-s + (−0.984 + 0.173i)17-s + (0.642 − 0.766i)19-s + (0.984 − 0.173i)20-s + (−0.342 + 0.939i)22-s + (−0.866 − 0.5i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $-0.994 + 0.103i$
Analytic conductor: \(0.515481\)
Root analytic conductor: \(0.515481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 111,\ (0:\ ),\ -0.994 + 0.103i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01507777497 - 0.2900766683i\)
\(L(\frac12)\) \(\approx\) \(0.01507777497 - 0.2900766683i\)
\(L(1)\) \(\approx\) \(0.4068582352 - 0.2822724154i\)
\(L(1)\) \(\approx\) \(0.4068582352 - 0.2822724154i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.642 - 0.766i)T \)
5 \( 1 + (-0.342 - 0.939i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.984 - 0.173i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
19 \( 1 + (0.642 - 0.766i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + iT \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.939 + 0.342i)T \)
59 \( 1 + (0.342 - 0.939i)T \)
61 \( 1 + (0.984 + 0.173i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 - T \)
79 \( 1 + (0.342 + 0.939i)T \)
83 \( 1 + (-0.173 - 0.984i)T \)
89 \( 1 + (-0.342 + 0.939i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−29.700649145192054606447731059890, −28.893702760981547089138565311184, −27.73612916134984131354771942384, −26.48774588088141392683378652950, −26.25106557496707588266909707115, −25.070316892333696901553504543089, −23.898887419641178401201557663002, −22.821217490301566497339851532927, −22.27255192167883975089279886488, −20.22965349520513738628761722324, −19.418694746921094644627299375348, −18.4323193703053877898518972256, −17.50617162344492105506232735526, −16.26002765892258620878722519320, −15.364136869277324447040971196539, −14.44579856505488840541153400577, −13.191274149942436354737976052260, −11.53997685318781237638422716805, −10.138352182476303718724916076424, −9.6016274913904059413542493718, −7.73982823038766218801322056271, −7.08764795115563492884391664867, −5.92933500442062114107848924283, −4.225573582377497463080316179962, −2.38315364131302752517338372697, 0.32556019236106231326480770868, 2.383755129879040363516691393131, 3.738105590091471430842410117579, 5.25468068476395412830199541223, 7.13021451537742118819921454481, 8.503332318542143610096841032691, 9.259068556715847493196695209433, 10.480045825698461906359604648653, 11.81118140338657241722218799993, 12.68165134355732942251813727748, 13.5547349774215534280240038264, 15.67245809560101443138921271041, 16.38718615701879383769275907561, 17.479664905292725609151510391854, 18.72123095233482104269121903461, 19.70066935682809863598541651625, 20.30097534533990485756432492588, 21.69060148520313347895733416887, 22.341923957209752149686621425325, 23.95967167482602851509452245286, 24.90764929990726916864899057641, 26.20302487993107473658863748944, 26.9406072938608732860067885405, 28.15358573252372581362243865832, 28.83590573348488075757038375374

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