Properties

Label 1-111-111.53-r1-0-0
Degree $1$
Conductor $111$
Sign $0.957 - 0.287i$
Analytic cond. $11.9286$
Root an. cond. $11.9286$
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  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)5-s + (−0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.173 + 0.984i)13-s + (0.5 + 0.866i)14-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + (0.766 − 0.642i)19-s + (−0.173 + 0.984i)20-s + (−0.939 + 0.342i)22-s + (0.5 + 0.866i)23-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)5-s + (−0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.173 + 0.984i)13-s + (0.5 + 0.866i)14-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + (0.766 − 0.642i)19-s + (−0.173 + 0.984i)20-s + (−0.939 + 0.342i)22-s + (0.5 + 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.957 - 0.287i$
Analytic conductor: \(11.9286\)
Root analytic conductor: \(11.9286\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 111,\ (1:\ ),\ 0.957 - 0.287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.304864998 - 0.1914219572i\)
\(L(\frac12)\) \(\approx\) \(1.304864998 - 0.1914219572i\)
\(L(1)\) \(\approx\) \(0.8880406127 - 0.1529740731i\)
\(L(1)\) \(\approx\) \(0.8880406127 - 0.1529740731i\)

Euler product

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

−28.94879753688154517035059609890, −28.27432239880924018758139814154, −27.163417513264332778990363212798, −25.99419647877618638445983522595, −25.03237608246890973498962531819, −24.826420055832463638947898301538, −23.02726803667579038142030484347, −22.35479769723105821943197207643, −20.64670463585705580613992354662, −19.8839388749292392226297261965, −18.49800598095819691876463184203, −17.76727121524475314783253934906, −16.68326003347233463692384424141, −15.79631766689422069726519942163, −14.60151075089655738151931379259, −13.42093551485875150964394765815, −12.18046127435677770153925421012, −10.34346641555326130101897997356, −9.62696747613512376133377648637, −8.68117221107272246151318143948, −7.114689688708487353771390115271, −6.09441058779221895418589572679, −4.99791956924835829126533760213, −2.62950112801522988345797309711, −0.950648236006412442306016949248, 1.09445858709821148517099860530, 2.651560646246027674242692983419, 3.87968757580469819455727117809, 6.1092767235142155704201765805, 7.08626895499615779695943898599, 8.82013142391115100866043740541, 9.61981927734682178265265680337, 10.648128754254216424527053574206, 11.751851793315518264423910736822, 13.2061174940305630904083978729, 13.89447074255545916965944813583, 15.796686799029053131336845636797, 16.894474304760878608921735220305, 17.612747123276404241416033408290, 19.02631468054669541727117909524, 19.44989014703712327867790041997, 20.95236151158587859467006305017, 21.74497520615915786860733513778, 22.57606571714349678979907902525, 24.24831942945990207967167277726, 25.4931293867433691961452734716, 26.2102795319482641912453190624, 26.95059805337091935968336944194, 28.49020370391185575454736414651, 29.0191973207066240162638761452

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