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Properties

Label 1-119-119.89-r1-0-0
Degree $1$
Conductor $119$
Sign $0.0125 - 0.999i$
Analytic cond. $12.7883$
Root an. cond. $12.7883$
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  + (0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s − i·6-s − 8-s + (0.5 + 0.866i)9-s + (0.866 + 0.5i)10-s + (−0.866 − 0.5i)11-s + (0.866 − 0.5i)12-s − 13-s − 15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + i·20-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s − i·6-s − 8-s + (0.5 + 0.866i)9-s + (0.866 + 0.5i)10-s + (−0.866 − 0.5i)11-s + (0.866 − 0.5i)12-s − 13-s − 15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0125 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0125 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $0.0125 - 0.999i$
Analytic conductor: \(12.7883\)
Root analytic conductor: \(12.7883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (1:\ ),\ 0.0125 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5390977936 - 0.5323690786i\)
\(L(\frac12)\) \(\approx\) \(0.5390977936 - 0.5323690786i\)
\(L(1)\) \(\approx\) \(0.8629306776 + 0.08801127480i\)
\(L(1)\) \(\approx\) \(0.8629306776 + 0.08801127480i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 - T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 - iT \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 - iT \)
43 \( 1 - T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.866 + 0.5i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 - iT \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + iT \)
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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

−29.32852352994478508488495955768, −28.395432685078672259708055910515, −27.3466360502849648224745261091, −26.41850552568678350619327914016, −24.95265826591282051386210702491, −23.546785475132513249653682015656, −22.8745041387734729371258481394, −21.78126201335404513713907990136, −21.33688898770155241765228061103, −20.20846423922868559994396445160, −18.69405189936084554088320464535, −17.92207008057654872851682699803, −16.877090952331693089361576451165, −15.27631632382868274758112749827, −14.45702832425211500549958789944, −13.04113253657651144860695166025, −12.20362320494196037454109524165, −10.830636977799984371001973453327, −10.2353344664844209892457594669, −9.29153904493803615677201539375, −6.94069414738572563205149576068, −5.596813378671643683519230813388, −4.85966682528575309462425404081, −3.23967777296623232279017196724, −1.767110132073562582086178043191, 0.28057690808344899328597376859, 2.45602213262898216197384625918, 4.71939496929002543361502803290, 5.45342054576510226874754801142, 6.51740646616702729867921458241, 7.64634473020185737666162001540, 8.99830498529230219534167364274, 10.49191355935487874278645651354, 12.01628456717477105330682762567, 13.032585872649971187933388903766, 13.613482898952261339168134476818, 15.09545750845779603388107673837, 16.377038719930329215943301038561, 17.11622709987852279658888128734, 17.85921202395113371960220841070, 19.00764587816758592605668593860, 20.88828175687849239172021163009, 21.7781430930872468270364280028, 22.59134321909181379376832423759, 23.85713350127712696468165499049, 24.33793587183253129717557589839, 25.293842913613816213676836341371, 26.39526874548868472598853288485, 27.60539008314541064986727693042, 28.84874511444154468782630494190

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