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Properties

Label 1-119-119.108-r0-0-0
Degree $1$
Conductor $119$
Sign $-0.0713 + 0.997i$
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  + (−0.965 + 0.258i)2-s + (0.793 + 0.608i)3-s + (0.866 − 0.5i)4-s + (−0.991 − 0.130i)5-s + (−0.923 − 0.382i)6-s + (−0.707 + 0.707i)8-s + (0.258 + 0.965i)9-s + (0.991 − 0.130i)10-s + (0.130 + 0.991i)11-s + (0.991 + 0.130i)12-s + i·13-s + (−0.707 − 0.707i)15-s + (0.5 − 0.866i)16-s + (−0.5 − 0.866i)18-s + (0.965 − 0.258i)19-s + (−0.923 + 0.382i)20-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.793 + 0.608i)3-s + (0.866 − 0.5i)4-s + (−0.991 − 0.130i)5-s + (−0.923 − 0.382i)6-s + (−0.707 + 0.707i)8-s + (0.258 + 0.965i)9-s + (0.991 − 0.130i)10-s + (0.130 + 0.991i)11-s + (0.991 + 0.130i)12-s + i·13-s + (−0.707 − 0.707i)15-s + (0.5 − 0.866i)16-s + (−0.5 − 0.866i)18-s + (0.965 − 0.258i)19-s + (−0.923 + 0.382i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0713 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0713 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $-0.0713 + 0.997i$
Analytic conductor: \(0.552633\)
Root analytic conductor: \(0.552633\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (0:\ ),\ -0.0713 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4951267042 + 0.5318105403i\)
\(L(\frac12)\) \(\approx\) \(0.4951267042 + 0.5318105403i\)
\(L(1)\) \(\approx\) \(0.6899323970 + 0.3353795698i\)
\(L(1)\) \(\approx\) \(0.6899323970 + 0.3353795698i\)

Euler product

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

−28.89658063062128614713407834824, −27.6138020165458738509415645418, −26.80851306892662534928031225482, −26.114607743125997431710130393874, −24.78795225532500638238144238641, −24.29555005416576221107600253432, −22.86587559162864487481712391319, −21.34076165679117855575886405454, −20.12572467827732231243273409924, −19.66309803377356959809862516596, −18.63795450653537831689618348821, −17.913391187506325463058585077300, −16.340868081960626918313657352145, −15.48998714871814172047316977708, −14.26187784325509520200835035552, −12.75864139811432739012308087178, −11.83425392594324267793638341745, −10.68820418125544644749117932660, −9.26918282745927556110812793510, −8.12323779737028498931978706412, −7.640675658421935268549346632, −6.23544825498695042963582002340, −3.677284947670670502513923902447, −2.75589075801952795210892531798, −0.90243495687237910624343506433, 1.91452019348737441147343256374, 3.55486516003972318036188014848, 4.940413189203700163937931200242, 7.00936607740722499657524584211, 7.83708398404989188084341418203, 8.98472861352680745592243650322, 9.79804886449151418993721281542, 11.09495813183395162589193222312, 12.16950497032075393985845615304, 14.06881327946060495186744176725, 15.11113028148837338536524507245, 15.87187407780925947284206584814, 16.71821595850917588184220308483, 18.18412931313489168886672780098, 19.27814278641791848620753242049, 20.022782022322673284110841954643, 20.73067233093361642054046778022, 22.17216486127496386029915326988, 23.64068521645688280336253280531, 24.51274566987994896727356042437, 25.77096633674278074607857798641, 26.30120272907496794925220023961, 27.43822181244991809728255940733, 27.891057396868815811900346350340, 29.07076653000384001786053886003

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