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Properties

Label 1-119-119.59-r1-0-0
Degree $1$
Conductor $119$
Sign $0.998 - 0.0590i$
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  + (0.866 + 0.5i)2-s + (0.258 − 0.965i)3-s + (0.5 + 0.866i)4-s + (0.965 − 0.258i)5-s + (0.707 − 0.707i)6-s + i·8-s + (−0.866 − 0.5i)9-s + (0.965 + 0.258i)10-s + (0.965 + 0.258i)11-s + (0.965 − 0.258i)12-s + 13-s − i·15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)18-s + (−0.866 − 0.5i)19-s + (0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.258 − 0.965i)3-s + (0.5 + 0.866i)4-s + (0.965 − 0.258i)5-s + (0.707 − 0.707i)6-s + i·8-s + (−0.866 − 0.5i)9-s + (0.965 + 0.258i)10-s + (0.965 + 0.258i)11-s + (0.965 − 0.258i)12-s + 13-s − i·15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)18-s + (−0.866 − 0.5i)19-s + (0.707 + 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $0.998 - 0.0590i$
Analytic conductor: \(12.7883\)
Root analytic conductor: \(12.7883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (1:\ ),\ 0.998 - 0.0590i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.802318553 - 0.1123720924i\)
\(L(\frac12)\) \(\approx\) \(3.802318553 - 0.1123720924i\)
\(L(1)\) \(\approx\) \(2.249922015 + 0.01623216431i\)
\(L(1)\) \(\approx\) \(2.249922015 + 0.01623216431i\)

Euler product

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

−28.94957905348786567382095993260, −28.032471254572960007633677297586, −27.042582252319064947215122728131, −25.544786448591587776329008713694, −25.095228016518249830640386935788, −23.48540169722834527554402419556, −22.47632427498947247247283733089, −21.61549298514984109089162130269, −21.05383118398341826301861977989, −19.99341465326220808863649552053, −18.8991628080248716450566505410, −17.34749077030737839873596098040, −16.15346103657557562082884745544, −15.02718521769007960706347601356, −14.10613346725756424340116207224, −13.403653109877277378084340669527, −11.78510280070205889850113020509, −10.70794646603880999468173880237, −9.86813501674525895398427633486, −8.74093621045659349462629139347, −6.464128784606674833106805852351, −5.58587414013420882282616172723, −4.19207712077848890037897358089, −3.148820570718655601170781698319, −1.66964190584675517191055878309, 1.50730351107540076491588590122, 2.82173523928539309258399240640, 4.48609168875075066949586434812, 6.16119781856090935375483819544, 6.52931804179828686321094884738, 8.11691171499340237101353818252, 9.13018481223440245253939651486, 11.10428734508036942819373408730, 12.38903253319277973226570177910, 13.172295221397501629675902723260, 14.04409980411712565298603421256, 14.87693767438535531381182450784, 16.497484763916757985216837930295, 17.38324301360664024745910190366, 18.31291123151834877830404535156, 19.86320303243607646138724839638, 20.77013703202387080386049288933, 21.86555812538144399943570225192, 22.935769522761471634705533235577, 23.91546815231340122513309875537, 24.854325136060993826988515909827, 25.45133946390120605838160475134, 26.25680758779544932958310020641, 28.13353689705530485483042961213, 29.2366490624898800479386476028

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