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Properties

Label 1-119-119.88-r1-0-0
Degree $1$
Conductor $119$
Sign $-0.493 + 0.870i$
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 + (−0.130 − 0.991i)3-s + (−0.866 + 0.5i)4-s + (−0.608 − 0.793i)5-s + (0.923 − 0.382i)6-s + (−0.707 − 0.707i)8-s + (−0.965 + 0.258i)9-s + (0.608 − 0.793i)10-s + (0.793 + 0.608i)11-s + (0.608 + 0.793i)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 + (0.923 + 0.382i)20-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.130 − 0.991i)3-s + (−0.866 + 0.5i)4-s + (−0.608 − 0.793i)5-s + (0.923 − 0.382i)6-s + (−0.707 − 0.707i)8-s + (−0.965 + 0.258i)9-s + (0.608 − 0.793i)10-s + (0.793 + 0.608i)11-s + (0.608 + 0.793i)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 + (0.923 + 0.382i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.493 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.493 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $-0.493 + 0.870i$
Analytic conductor: \(12.7883\)
Root analytic conductor: \(12.7883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (1:\ ),\ -0.493 + 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4691835840 + 0.8051596605i\)
\(L(\frac12)\) \(\approx\) \(0.4691835840 + 0.8051596605i\)
\(L(1)\) \(\approx\) \(0.8148194873 + 0.2765215393i\)
\(L(1)\) \(\approx\) \(0.8148194873 + 0.2765215393i\)

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 \( 1 + (-0.130 - 0.991i)T \)
5 \( 1 + (-0.608 - 0.793i)T \)
11 \( 1 + (0.793 + 0.608i)T \)
13 \( 1 + iT \)
19 \( 1 + (0.258 + 0.965i)T \)
23 \( 1 + (-0.130 + 0.991i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + (0.130 + 0.991i)T \)
37 \( 1 + (-0.793 + 0.608i)T \)
41 \( 1 + (-0.382 - 0.923i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.965 - 0.258i)T \)
59 \( 1 + (-0.258 + 0.965i)T \)
61 \( 1 + (-0.991 - 0.130i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.923 + 0.382i)T \)
73 \( 1 + (0.991 - 0.130i)T \)
79 \( 1 + (0.130 - 0.991i)T \)
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

−28.40233059298680965799191000296, −27.516391093998311483476905175581, −26.93330452660386615719437397064, −25.95494118671907643788174927175, −24.18878310191629300503203938192, −22.87399729799347036352483747204, −22.35340703919714358718419176984, −21.55805200487683379077994599129, −20.27487786536301582769660685801, −19.62370164850417945846781488105, −18.40942108972982129877501854194, −17.26668666376912084682642929459, −15.77886630702503963900163040368, −14.80859603398610935528072599716, −13.96439430408516715359021527489, −12.34900046866597721016915601698, −11.21251869512021702637450855399, −10.67545971511445293152600177947, −9.47388807311365808111493826930, −8.29774513724208736805584208678, −6.27781926053040303097793986964, −4.86755936232801884692439591697, −3.66710216751008427695763262963, −2.83387844119186945287131454975, −0.39170061522081987887351159836, 1.40771712274125270519051898770, 3.76464532277887399660411543023, 5.04394283887115277104993915623, 6.37338728297440255722707301061, 7.380594355310606876924524268984, 8.35390606398681602013737975739, 9.414113050555924264046852647677, 11.80009571538170459164216888219, 12.316554675779686868272672793817, 13.550273718666920370901101440022, 14.437774786462037234495904539424, 15.78070509990780491142483724074, 16.831975484003173215775208421441, 17.54275636211342441647831588073, 18.83609725160024150193449422136, 19.72117473889198621933341235193, 21.13269475676686673252137980699, 22.630419044705983979634705913470, 23.34306730928094051258941763410, 24.22554608227404749603313326008, 24.91103252325459984974903289834, 25.82699630625485931250009445676, 27.168981410778430084988763216, 28.08337460092224673609764965468, 29.20779688914248927960705461843

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