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Properties

Label 1-119-119.27-r0-0-0
Degree $1$
Conductor $119$
Sign $0.855 + 0.518i$
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 + (−0.382 − 0.923i)3-s + i·4-s + (−0.923 + 0.382i)5-s + (−0.382 + 0.923i)6-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.923 + 0.382i)10-s + (−0.382 + 0.923i)11-s + (0.923 − 0.382i)12-s + i·13-s + (0.707 + 0.707i)15-s − 16-s + 18-s + (0.707 + 0.707i)19-s + (−0.382 − 0.923i)20-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.382 − 0.923i)3-s + i·4-s + (−0.923 + 0.382i)5-s + (−0.382 + 0.923i)6-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.923 + 0.382i)10-s + (−0.382 + 0.923i)11-s + (0.923 − 0.382i)12-s + i·13-s + (0.707 + 0.707i)15-s − 16-s + 18-s + (0.707 + 0.707i)19-s + (−0.382 − 0.923i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $0.855 + 0.518i$
Analytic conductor: \(0.552633\)
Root analytic conductor: \(0.552633\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (0:\ ),\ 0.855 + 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3675865500 + 0.1027029964i\)
\(L(\frac12)\) \(\approx\) \(0.3675865500 + 0.1027029964i\)
\(L(1)\) \(\approx\) \(0.4951280362 - 0.1027655335i\)
\(L(1)\) \(\approx\) \(0.4951280362 - 0.1027655335i\)

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 \( 1 + (-0.382 - 0.923i)T \)
5 \( 1 + (-0.923 + 0.382i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 + iT \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (-0.923 + 0.382i)T \)
31 \( 1 + (0.382 + 0.923i)T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (-0.923 - 0.382i)T \)
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 \( 1 + (0.923 + 0.382i)T \)
67 \( 1 - T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (-0.923 + 0.382i)T \)
79 \( 1 + (-0.382 + 0.923i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 - iT \)
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

−28.48514380411476271458718353518, −27.89335178033690831398388121044, −26.9250474282707766096648352664, −26.445022899365811774123763263709, −25.05192698353751348700974332537, −23.94196140138998714519360295109, −23.17122577338976277124990087190, −22.10190018733988032469910026233, −20.61249935459445662719779773211, −19.802795746747025669755733500231, −18.61163036944261965311132141976, −17.373685646925932988580014852177, −16.4741647584706582148133687830, −15.598549812482424334016858670786, −15.05261760628004574211742608777, −13.43252405635868159533833914157, −11.593886172369132636483092260871, −10.83196744141778593836434660313, −9.57422888024761500405113955814, −8.52391042307683468350334721355, −7.50019120734174128916230020084, −5.81501770691643565660687282342, −4.93479000694645055805169501130, −3.382713380232792049578559507117, −0.49704163090793426628833281133, 1.56984813765412255150753974387, 2.987091451366423067878431150332, 4.592442848760472105927905349038, 6.75407518663508475153951547056, 7.52462513230418616832497478265, 8.576912757131475144357529114832, 10.15169311311800863371255606335, 11.33333987289142316393342516990, 12.04977787026687402146094316911, 12.95281471854414489473478042594, 14.41681166292864370233332686087, 16.04438755914367084505747538079, 17.01390145090071794659965621316, 18.28474428690323533367548650791, 18.75516083979073148332734574718, 19.77121146292694765907326806085, 20.684719830953477148150244612582, 22.24348587410139601898631573719, 23.01810428535678081535722385381, 24.09310233642850850728414499257, 25.35429220201119490792519715158, 26.34407898243489206571530929736, 27.33351825392154785646574493259, 28.45112584206253485000289599552, 28.99242979028729637695820899967

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