↑

Properties

Label 1-119-119.16-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

Related objects

Downloads

Learn more

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: $\chi_{119} (16, \cdot )$
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 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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