↑

Properties

Label 1-199-199.125-r0-0-0
Degree $1$
Conductor $199$
Sign $0.979 - 0.200i$
Analytic cond. $0.924152$
Root an. cond. $0.924152$
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  + (−0.654 − 0.755i)2-s + (0.841 + 0.540i)3-s + (−0.142 + 0.989i)4-s + (−0.959 − 0.281i)5-s + (−0.142 − 0.989i)6-s + (0.841 − 0.540i)7-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (0.415 + 0.909i)10-s + (0.415 + 0.909i)11-s + (−0.654 + 0.755i)12-s + (−0.654 − 0.755i)13-s + (−0.959 − 0.281i)14-s + (−0.654 − 0.755i)15-s + (−0.959 − 0.281i)16-s + (0.415 + 0.909i)17-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (0.841 + 0.540i)3-s + (−0.142 + 0.989i)4-s + (−0.959 − 0.281i)5-s + (−0.142 − 0.989i)6-s + (0.841 − 0.540i)7-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (0.415 + 0.909i)10-s + (0.415 + 0.909i)11-s + (−0.654 + 0.755i)12-s + (−0.654 − 0.755i)13-s + (−0.959 − 0.281i)14-s + (−0.654 − 0.755i)15-s + (−0.959 − 0.281i)16-s + (0.415 + 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.979 - 0.200i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.979 - 0.200i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.042079991 - 0.1054490327i\)
\(L(\frac12)\) \(\approx\) \(1.042079991 - 0.1054490327i\)
\(L(1)\) \(\approx\) \(0.9584254925 - 0.1240671064i\)
\(L(1)\) \(\approx\) \(0.9584254925 - 0.1240671064i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.654 - 0.755i)T \)
3 \( 1 + (0.841 + 0.540i)T \)
5 \( 1 + (-0.959 - 0.281i)T \)
7 \( 1 + (0.841 - 0.540i)T \)
11 \( 1 + (0.415 + 0.909i)T \)
13 \( 1 + (-0.654 - 0.755i)T \)
17 \( 1 + (0.415 + 0.909i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.142 - 0.989i)T \)
29 \( 1 + (-0.654 + 0.755i)T \)
31 \( 1 + (0.841 - 0.540i)T \)
37 \( 1 + T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + T \)
47 \( 1 + (0.841 - 0.540i)T \)
53 \( 1 + (-0.654 + 0.755i)T \)
59 \( 1 + (-0.142 - 0.989i)T \)
61 \( 1 + (-0.959 + 0.281i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (-0.959 - 0.281i)T \)
83 \( 1 + (-0.654 - 0.755i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (0.841 - 0.540i)T \)
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

−27.05498318571215485209256233894, −26.076756909122634435882561790597, −24.91603272622533553054544108139, −24.31370461257175267193241100798, −23.718600344653075354303554014687, −22.44504164557545543584961734633, −21.023522012808863855110683060947, −19.87281847161020399078979005331, −19.08158455783866284977130742906, −18.542280096980444502666182545318, −17.500282968453471923361607890768, −16.15292164490719376307548397257, −15.34824003269325808358739632227, −14.35215147028769456608660816727, −13.87371519109529878432828702347, −11.97111361677390432668274322860, −11.31145405092474648025802186106, −9.567862319111052844618610475631, −8.76754420889957700240248457549, −7.727510798187766783469253145132, −7.25187841023588324324542011499, −5.80600235051088603660935156103, −4.322277145052351715047787789642, −2.74407205758508903144475130118, −1.181110827476345765778688117815, 1.359991937410237863428418351079, 2.8361326728081287169761327247, 4.050219691726837743951698005102, 4.70870568067930942677678073427, 7.53291656218233337076260906888, 7.819807771765518430612129302072, 8.9599079208437446864108631725, 10.04086609061472201410043883747, 10.88507076950463756710184697050, 12.04684204697514096642358385221, 12.95466278831419942050187616401, 14.39527855622058465822081571857, 15.20422130293792711327941168445, 16.42984020980538777566291389110, 17.29296932315678049586453140402, 18.4933708778419070283407502325, 19.61337735812797307245795362172, 20.24232653493685207509352038389, 20.66719738163468733374359594395, 21.93443328168130303608982896589, 22.85035878771787579568973081199, 24.28996432304382319653848633518, 25.18735982990706807799028627534, 26.41854428513672930045203260195, 26.91265691452861317448257785611

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