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Properties

Label 1-119-119.6-r0-0-0
Degree $1$
Conductor $119$
Sign $-0.837 + 0.547i$
Analytic cond. $0.552633$
Root an. cond. $0.552633$
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  + (0.707 + 0.707i)2-s + (−0.923 + 0.382i)3-s + i·4-s + (0.382 + 0.923i)5-s + (−0.923 − 0.382i)6-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (−0.382 + 0.923i)10-s + (−0.923 − 0.382i)11-s + (−0.382 − 0.923i)12-s + i·13-s + (−0.707 − 0.707i)15-s − 16-s + 18-s + (−0.707 − 0.707i)19-s + (−0.923 + 0.382i)20-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.923 + 0.382i)3-s + i·4-s + (0.382 + 0.923i)5-s + (−0.923 − 0.382i)6-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (−0.382 + 0.923i)10-s + (−0.923 − 0.382i)11-s + (−0.382 − 0.923i)12-s + i·13-s + (−0.707 − 0.707i)15-s − 16-s + 18-s + (−0.707 − 0.707i)19-s + (−0.923 + 0.382i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $-0.837 + 0.547i$
Analytic conductor: \(0.552633\)
Root analytic conductor: \(0.552633\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (0:\ ),\ -0.837 + 0.547i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3051174217 + 1.024455680i\)
\(L(\frac12)\) \(\approx\) \(0.3051174217 + 1.024455680i\)
\(L(1)\) \(\approx\) \(0.7696574886 + 0.7856176571i\)
\(L(1)\) \(\approx\) \(0.7696574886 + 0.7856176571i\)

Euler product

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

−28.79070472558517166282218748921, −28.17389707179300093345074642076, −27.213036997454679282010474818889, −25.17339117449492446344633289800, −24.47603604057743674868557671008, −23.31720473672543548342433352806, −22.837610483617573696785639800041, −21.44281856868729394940114383641, −20.821321678635067728863531651683, −19.60799540699056778020619013803, −18.4045168506845615608586417952, −17.459277861528155450055261375385, −16.22254030658435231997424240109, −15.08511929331746285229238364334, −13.378010136822308845406248022362, −12.855125191271579429324484962544, −11.99099827843419773070615786398, −10.672261055483333086470689326522, −9.86756509188240645822975513780, −8.0750664319442218718092730087, −6.30869555046806777906701195417, −5.32859378311603003478821801512, −4.484942650508736821385135526000, −2.435695877007640825270315520288, −0.94226626654568488407049412610, 2.710743645300153955966141937101, 4.21344285000112689542449543791, 5.42593378975691107593390805951, 6.43469033814034479184759804130, 7.31709464817159516905941144260, 9.103979924250952098831172687299, 10.64576823112022882195086097694, 11.4653915106826299222953840908, 12.80898724411866776705321553768, 13.89858471466087487461495842611, 15.066930211938604178167683393464, 15.910758678693970966186168870503, 17.00322562398669996567434020770, 17.86648716236631204309483531110, 18.924055770431924988909492549204, 21.0958913237535286996821140067, 21.54588568541872962272474609959, 22.510506411295029000170482782884, 23.42556525777692757799304970331, 24.108451744570359839141313886838, 25.61795311758925163699546443217, 26.39392496602285169255548171621, 27.221225329299839092866179294211, 28.791049900298096151263146058586, 29.541148792128097610381720385191

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