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Properties

Label 1-119-119.67-r0-0-0
Degree $1$
Conductor $119$
Sign $0.0633 + 0.997i$
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  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s − 6-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)12-s + 13-s + 15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s − 20-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s − 6-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)12-s + 13-s + 15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s − 20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(0.552633\)
Root analytic conductor: \(0.552633\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (0:\ ),\ 0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7395766865 + 0.6941307269i\)
\(L(\frac12)\) \(\approx\) \(0.7395766865 + 0.6941307269i\)
\(L(1)\) \(\approx\) \(0.8602188261 + 0.5207597101i\)
\(L(1)\) \(\approx\) \(0.8602188261 + 0.5207597101i\)

Euler product

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

−29.21352693684699365352288523441, −28.01037664494439832085969731531, −26.753955652404654590962983548852, −25.92378842199122893546869726636, −25.23156283638037771000134262002, −23.79134498427505655352792782019, −22.58872532040404753402337511559, −21.56434928043545891462760609513, −20.57773908508357004485497349251, −19.34356073672358427422599908252, −18.77270954268158014243800505459, −17.88252760388448585100273782276, −16.92037394095974543375936708780, −15.05949969492908467145742915545, −13.636842044796262635610362793580, −13.31353398626273478158716649955, −11.66858642817689055918792259580, −10.93507412116790114210820391274, −9.45012958284800247737613853050, −8.52714700537861474363421792085, −7.25912092777400212022992478753, −6.07270750520787274395236668747, −3.66589884283648345597505083171, −2.67987721724846746448847029356, −1.34778154864226548141337533994, 1.72343709506298242277298864133, 4.06336415794236928222194805320, 5.088571447260442331761534517238, 6.32515620702562396511474275336, 8.0228119815987259708981482976, 8.92780775820332825082932802594, 9.70984004502317080397096776431, 10.80052495940216961686188340466, 12.7801492578387912917980449755, 14.0004149386233232042917699408, 14.87730353189799850592534791315, 16.007377636804515270349978839491, 16.73164061981212339925836497502, 17.69244611453500009603298970502, 19.059796090544968681791214013928, 20.26242571428869999097653325650, 20.965336878504876807103651482634, 22.38013802270856559447817090866, 23.36924727941291260403273151209, 24.79455045854034238095662090382, 25.33802470099591904149056013170, 26.18788680171413776400032699257, 27.376956126624977926167258108268, 28.055792454792376810687798141652, 28.833312597842647303656157819745

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