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Properties

Label 1-193-193.158-r1-0-0
Degree $1$
Conductor $193$
Sign $-0.461 - 0.887i$
Analytic cond. $20.7407$
Root an. cond. $20.7407$
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  + (−0.831 − 0.555i)2-s + (0.923 − 0.382i)3-s + (0.382 + 0.923i)4-s + (0.956 + 0.290i)5-s + (−0.980 − 0.195i)6-s + (0.707 − 0.707i)7-s + (0.195 − 0.980i)8-s + (0.707 − 0.707i)9-s + (−0.634 − 0.773i)10-s + (−0.881 − 0.471i)11-s + (0.707 + 0.707i)12-s + (−0.773 − 0.634i)13-s + (−0.980 + 0.195i)14-s + (0.995 − 0.0980i)15-s + (−0.707 + 0.707i)16-s + (−0.956 + 0.290i)17-s + ⋯
L(s)  = 1  + (−0.831 − 0.555i)2-s + (0.923 − 0.382i)3-s + (0.382 + 0.923i)4-s + (0.956 + 0.290i)5-s + (−0.980 − 0.195i)6-s + (0.707 − 0.707i)7-s + (0.195 − 0.980i)8-s + (0.707 − 0.707i)9-s + (−0.634 − 0.773i)10-s + (−0.881 − 0.471i)11-s + (0.707 + 0.707i)12-s + (−0.773 − 0.634i)13-s + (−0.980 + 0.195i)14-s + (0.995 − 0.0980i)15-s + (−0.707 + 0.707i)16-s + (−0.956 + 0.290i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.461 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.461 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $-0.461 - 0.887i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ -0.461 - 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9836422782 - 1.620974234i\)
\(L(\frac12)\) \(\approx\) \(0.9836422782 - 1.620974234i\)
\(L(1)\) \(\approx\) \(1.007495341 - 0.6028300774i\)
\(L(1)\) \(\approx\) \(1.007495341 - 0.6028300774i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad193 \( 1 \)
good2 \( 1 + (-0.831 - 0.555i)T \)
3 \( 1 + (0.923 - 0.382i)T \)
5 \( 1 + (0.956 + 0.290i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (-0.881 - 0.471i)T \)
13 \( 1 + (-0.773 - 0.634i)T \)
17 \( 1 + (-0.956 + 0.290i)T \)
19 \( 1 + (0.290 - 0.956i)T \)
23 \( 1 + (-0.831 - 0.555i)T \)
29 \( 1 + (0.0980 - 0.995i)T \)
31 \( 1 + (0.555 - 0.831i)T \)
37 \( 1 + (0.0980 + 0.995i)T \)
41 \( 1 + (0.881 + 0.471i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (-0.634 + 0.773i)T \)
53 \( 1 + (-0.773 - 0.634i)T \)
59 \( 1 + (0.923 - 0.382i)T \)
61 \( 1 + (0.290 + 0.956i)T \)
67 \( 1 + (0.555 - 0.831i)T \)
71 \( 1 + (-0.290 + 0.956i)T \)
73 \( 1 + (-0.0980 + 0.995i)T \)
79 \( 1 + (0.471 - 0.881i)T \)
83 \( 1 + (-0.195 - 0.980i)T \)
89 \( 1 + (0.773 - 0.634i)T \)
97 \( 1 + (-0.831 + 0.555i)T \)
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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

−26.882059763628157622858422897298, −26.16699379028544153025609862741, −25.2372873172840734438101709588, −24.69025254101329970613993046955, −23.854225001187602715287807256748, −22.1048897842628798510518717529, −21.13589271903536199943012390326, −20.438802062680674273259599203458, −19.3814530376565490206583813149, −18.22798611826604377295475910577, −17.69889576319928834441541703441, −16.3413995509414450557798248154, −15.54886882209890502653539865384, −14.4774431228733723192019960934, −13.90269829708816733040820368310, −12.398162875592194285826710976578, −10.769361813430441565577704665612, −9.80009915717342519572228579875, −9.07740657386640855873579262088, −8.17615832465547377772549265, −7.11704555699770306062664253480, −5.5577083984028894023590855249, −4.71902015265811565541374907044, −2.39834120328313444123401458409, −1.79225919631185230442506848734, 0.73599115478638206752600319946, 2.17120878947742185192138990392, 2.83878140384693991484726433853, 4.48035172123392693793580590009, 6.440490829691871431211575794641, 7.624578265573942829384408802962, 8.31206503535668233529453058919, 9.58074508458073249754053687914, 10.32130998173061371015336023153, 11.37610911079479557021011002694, 12.98311010719430164153765016629, 13.46782108973462455107793762474, 14.65331246105330798793169919454, 15.854783342974668923240166952271, 17.44149348696155806402209321785, 17.77459806412808960707812394912, 18.81197733231765069770396603935, 19.84073298389227532396938586383, 20.619387919780295867002601408329, 21.32748077168762953706098603719, 22.33504818595055700139328110528, 24.22931218678197412886695445876, 24.62538088221439288866518825979, 26.024086833404096359948588302559, 26.32087370787715906843845862368

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