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Properties

Label 1-199-199.121-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

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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: $\chi_{199} (121, \cdot )$
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 \( 1 + (0.841 + 0.540i)T \)
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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

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

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