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Properties

Label 1-193-193.125-r1-0-0
Degree $1$
Conductor $193$
Sign $-0.770 + 0.637i$
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  + (−0.980 − 0.195i)2-s + (−0.382 + 0.923i)3-s + (0.923 + 0.382i)4-s + (0.995 + 0.0980i)5-s + (0.555 − 0.831i)6-s + (−0.707 − 0.707i)7-s + (−0.831 − 0.555i)8-s + (−0.707 − 0.707i)9-s + (−0.956 − 0.290i)10-s + (0.634 − 0.773i)11-s + (−0.707 + 0.707i)12-s + (0.290 + 0.956i)13-s + (0.555 + 0.831i)14-s + (−0.471 + 0.881i)15-s + (0.707 + 0.707i)16-s + (−0.995 + 0.0980i)17-s + ⋯
L(s)  = 1  + (−0.980 − 0.195i)2-s + (−0.382 + 0.923i)3-s + (0.923 + 0.382i)4-s + (0.995 + 0.0980i)5-s + (0.555 − 0.831i)6-s + (−0.707 − 0.707i)7-s + (−0.831 − 0.555i)8-s + (−0.707 − 0.707i)9-s + (−0.956 − 0.290i)10-s + (0.634 − 0.773i)11-s + (−0.707 + 0.707i)12-s + (0.290 + 0.956i)13-s + (0.555 + 0.831i)14-s + (−0.471 + 0.881i)15-s + (0.707 + 0.707i)16-s + (−0.995 + 0.0980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $-0.770 + 0.637i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ -0.770 + 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2033822511 + 0.5645485137i\)
\(L(\frac12)\) \(\approx\) \(0.2033822511 + 0.5645485137i\)
\(L(1)\) \(\approx\) \(0.5867373320 + 0.1789686842i\)
\(L(1)\) \(\approx\) \(0.5867373320 + 0.1789686842i\)

Euler product

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

−26.05810120580421225732773830419, −25.428616694617820304143244842845, −24.8726935092598693037417040610, −23.9738725985243500467938183119, −22.60970545777688210789327014851, −21.81774836827095639621937286205, −20.20405036809422659680886696001, −19.6595260480361720694966147018, −18.36436751711261648799851417349, −17.853773570410740694067378039254, −17.15529735515429712677856510503, −15.992077824273923675213680233539, −14.90268402064492256107239318700, −13.48105768368260913558461408098, −12.5797817069488823915063196555, −11.53662051912043128404140512747, −10.3048277457741289562698457720, −9.30629526523327682162425157401, −8.38766242264321887703539122978, −6.93866838126156874236086363493, −6.30316227383307201349290762420, −5.34288642486381614973689922166, −2.65069611504646051670567548281, −1.794791282265032338497181997640, −0.3018492107949454590961095704, 1.36420831254100622073643655752, 3.03184544256131687156642200175, 4.221317461787987095540504876487, 6.19943619697577511034732440757, 6.512475832916919539302821591908, 8.47736733842600905897162956375, 9.421242784547125610742111596036, 10.12126886755636932594437373727, 10.95315072779265107229321013639, 11.99791926561601151187796327415, 13.525419057554691492378594988786, 14.59356869744884964196905768325, 16.13780495296486962345708211559, 16.53831390782570945900427324359, 17.38770182718181818186567993681, 18.367503305339702004642223238, 19.59690047173385157829259481199, 20.436310169556467247728286488872, 21.53083846871614517554345062881, 21.96386214217800054461775788774, 23.31136460431382598945384709477, 24.644311613644412343095226206882, 25.6510118872014876173628340068, 26.50973082672585445528052402547, 26.915376901899033353065801910578

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