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Properties

Label 1-143-143.32-r0-0-0
Degree $1$
Conductor $143$
Sign $-0.533 - 0.846i$
Analytic cond. $0.664089$
Root an. cond. $0.664089$
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)2-s + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s − i·5-s + (−0.866 − 0.5i)6-s + (0.866 + 0.5i)7-s − i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s − 12-s + 14-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + i·18-s + (−0.866 − 0.5i)19-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s − i·5-s + (−0.866 − 0.5i)6-s + (0.866 + 0.5i)7-s − i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s − 12-s + 14-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + i·18-s + (−0.866 − 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.533 - 0.846i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ -0.533 - 0.846i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7511986968 - 1.361142335i\)
\(L(\frac12)\) \(\approx\) \(0.7511986968 - 1.361142335i\)
\(L(1)\) \(\approx\) \(1.105668895 - 0.9566532758i\)
\(L(1)\) \(\approx\) \(1.105668895 - 0.9566532758i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 - iT \)
7 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + iT \)
53 \( 1 + T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 + iT \)
89 \( 1 + (0.866 - 0.5i)T \)
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

−28.80847192678331121106385368201, −27.13897244667800716813947648431, −26.83166017257969484043713835888, −25.720851991339988460641956987914, −24.57375998668792028599808100287, −23.229398079095347072682702661532, −22.92881893427161708641231295082, −21.74939903932029855461743872538, −21.11416200325825154549775058858, −20.0781019789816287464151228408, −18.23489829259344682511690577033, −17.31932860060896554460203851730, −16.37798032535851026592052153378, −15.19647649952801421493145151249, −14.61210457084724168150945290800, −13.6369525380797279199890079836, −11.98717683630425313685162862720, −11.12600531294466617343083647107, −10.311207932740239631931394671137, −8.52003963956287163504501285977, −7.14469140563633673048930345747, −6.16900348808263440713445034, −4.85769045029722602065852481395, −3.984260005985334222826248846422, −2.62402277341173336311851569492, 1.270278108878699823518084427898, 2.30440222746446904192245962956, 4.36622109584558286458950090450, 5.31253812293272435097544973000, 6.27461018394639198914792996883, 7.80249596635769350576936093543, 9.05336071604074884867907321938, 10.85559530387101596575883746527, 11.625599716459771061605052190898, 12.65828426369246167922461082586, 13.2145025373012197428517371441, 14.48317917298045830735248191609, 15.631778674873224776175111463379, 16.96703982305744332135206367002, 17.93250991932621089960051760753, 19.20665651025219210423100566114, 19.97785792378785963379601052071, 21.20112246556485774518827309134, 21.88494096739672544101612036502, 23.274005296383212056116858819946, 23.95126100904154405283675605829, 24.5741469655824578227122107351, 25.473105186213034793779913817968, 27.66333119465544432054492651082, 28.13743862973901199994931414561

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