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Properties

Label 1-119-119.39-r1-0-0
Degree $1$
Conductor $119$
Sign $-0.225 - 0.974i$
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  + (−0.258 − 0.965i)2-s + (0.991 − 0.130i)3-s + (−0.866 + 0.5i)4-s + (0.793 − 0.608i)5-s + (−0.382 − 0.923i)6-s + (0.707 + 0.707i)8-s + (0.965 − 0.258i)9-s + (−0.793 − 0.608i)10-s + (0.608 − 0.793i)11-s + (−0.793 + 0.608i)12-s + i·13-s + (0.707 − 0.707i)15-s + (0.5 − 0.866i)16-s + (−0.5 − 0.866i)18-s + (−0.258 − 0.965i)19-s + (−0.382 + 0.923i)20-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (0.991 − 0.130i)3-s + (−0.866 + 0.5i)4-s + (0.793 − 0.608i)5-s + (−0.382 − 0.923i)6-s + (0.707 + 0.707i)8-s + (0.965 − 0.258i)9-s + (−0.793 − 0.608i)10-s + (0.608 − 0.793i)11-s + (−0.793 + 0.608i)12-s + i·13-s + (0.707 − 0.707i)15-s + (0.5 − 0.866i)16-s + (−0.5 − 0.866i)18-s + (−0.258 − 0.965i)19-s + (−0.382 + 0.923i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $-0.225 - 0.974i$
Analytic conductor: \(12.7883\)
Root analytic conductor: \(12.7883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (1:\ ),\ -0.225 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.483265398 - 1.866674439i\)
\(L(\frac12)\) \(\approx\) \(1.483265398 - 1.866674439i\)
\(L(1)\) \(\approx\) \(1.214081013 - 0.8432104931i\)
\(L(1)\) \(\approx\) \(1.214081013 - 0.8432104931i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (0.991 - 0.130i)T \)
5 \( 1 + (0.793 - 0.608i)T \)
11 \( 1 + (0.608 - 0.793i)T \)
13 \( 1 + iT \)
19 \( 1 + (-0.258 - 0.965i)T \)
23 \( 1 + (0.991 + 0.130i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 + (-0.991 + 0.130i)T \)
37 \( 1 + (-0.608 - 0.793i)T \)
41 \( 1 + (-0.923 + 0.382i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.965 + 0.258i)T \)
59 \( 1 + (0.258 - 0.965i)T \)
61 \( 1 + (-0.130 + 0.991i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.382 + 0.923i)T \)
73 \( 1 + (0.130 + 0.991i)T \)
79 \( 1 + (-0.991 - 0.130i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.923 + 0.382i)T \)
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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

−29.1950833370495878655847298102, −27.64234371831753463098288182031, −26.9864628211382598374521401634, −25.82448990859460349391757126692, −25.29112159792188229091906770095, −24.642757450176559362523851951761, −23.065799783086659304326800542433, −22.22297606680319633807617068053, −20.99409704119822077329845960096, −19.76630421695744437166960192564, −18.67860666658343085521763780588, −17.80253777921432857156093451735, −16.730702683980873751941272631464, −15.23466370955151568002831266773, −14.74518822294305557413392122876, −13.755408948542888177438073589436, −12.76149081058950250854290388201, −10.40186149272452179532429637107, −9.71791843744150836523579239402, −8.58831306529615990806890090699, −7.4209877904138599971861561009, −6.40128353484051748759701272665, −4.92910128314827657746475862006, −3.370042957051767265435378476797, −1.65407866589513028391805883117, 1.12447605245480824909521371182, 2.25906778262695269920852603274, 3.608346824971820732572987709974, 4.92746564605804716232209571155, 6.894771127347114405516804066965, 8.742411789880329852636793625132, 8.95496611814468944694838839721, 10.18694081952406863687482066396, 11.597889702305038053038166922180, 12.89077308942057555695063259222, 13.644547792616944758860867965630, 14.48985283244564122190046604096, 16.32546008524707026971668701337, 17.389378540213539948158755073947, 18.58278741021318373363145958215, 19.48426238997443170875735544477, 20.300391051731453516441468252865, 21.48967069750885966335875953637, 21.70413490816358280429986564197, 23.61042693763526256419313862383, 24.717278019410623169650403482, 25.7266444787959404613286806753, 26.623065733998024163430498749750, 27.592733136664840974737607910917, 28.80986382805727912076158297835

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