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Properties

Label 1-193-193.20-r1-0-0
Degree $1$
Conductor $193$
Sign $0.623 - 0.782i$
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  + (0.195 + 0.980i)2-s + (0.382 + 0.923i)3-s + (−0.923 + 0.382i)4-s + (−0.634 + 0.773i)5-s + (−0.831 + 0.555i)6-s + (−0.707 + 0.707i)7-s + (−0.555 − 0.831i)8-s + (−0.707 + 0.707i)9-s + (−0.881 − 0.471i)10-s + (0.0980 − 0.995i)11-s + (−0.707 − 0.707i)12-s + (−0.471 − 0.881i)13-s + (−0.831 − 0.555i)14-s + (−0.956 − 0.290i)15-s + (0.707 − 0.707i)16-s + (0.634 + 0.773i)17-s + ⋯
L(s)  = 1  + (0.195 + 0.980i)2-s + (0.382 + 0.923i)3-s + (−0.923 + 0.382i)4-s + (−0.634 + 0.773i)5-s + (−0.831 + 0.555i)6-s + (−0.707 + 0.707i)7-s + (−0.555 − 0.831i)8-s + (−0.707 + 0.707i)9-s + (−0.881 − 0.471i)10-s + (0.0980 − 0.995i)11-s + (−0.707 − 0.707i)12-s + (−0.471 − 0.881i)13-s + (−0.831 − 0.555i)14-s + (−0.956 − 0.290i)15-s + (0.707 − 0.707i)16-s + (0.634 + 0.773i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.623 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.623 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $0.623 - 0.782i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ 0.623 - 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4250062101 + 0.2048140768i\)
\(L(\frac12)\) \(\approx\) \(-0.4250062101 + 0.2048140768i\)
\(L(1)\) \(\approx\) \(0.3917975778 + 0.7050198852i\)
\(L(1)\) \(\approx\) \(0.3917975778 + 0.7050198852i\)

Euler product

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

−26.15953186785737641233844165961, −24.87197794599962052953505395691, −23.83733437199961204810935226044, −23.20555276327165452888519227737, −22.3582460687068570466876419460, −20.6845964666645843636330315643, −20.24421303306749503101517131620, −19.4412019998108813603997766564, −18.68960306068743439469807598870, −17.49752990566683621548484946334, −16.47881417726309408563208343869, −14.90931713135423166093389909092, −13.84516379074241262693004079068, −13.01653327344440393677059574897, −12.19897359770921263257611770094, −11.52275035698351791857762778386, −9.82469818142747536501465795095, −9.11770623291577909736499966583, −7.78545931539664737222658162803, −6.78024754458156231732415410856, −4.99350658019211998672679610075, −3.91767125688644583013746295602, −2.66593792041859242937012299960, −1.26460750894296576089580850783, −0.159521108856181204606122544290, 3.25137692925504024004246335583, 3.478409975654738982927377875026, 5.24412007147647505634855673714, 6.054534068239987485472638527201, 7.542279417179362581422011132606, 8.40787276928123247425645438370, 9.500511121853854251577508841689, 10.509184733107185918029192818987, 11.90365143670906077295998861606, 13.17978918755699444114948278499, 14.48940175998725447210911522681, 14.94286824690654132131626332384, 15.994239409044864321889026544921, 16.43706970196738324064191401460, 17.915764238609895524261813718627, 19.00374103339076494171341700496, 19.75529566250359526839524249190, 21.442149181048062091125216661054, 22.06626298606192698412907036590, 22.742139879386093931995060510096, 23.77299970685322696153212734267, 25.1296894615299657795651547590, 25.6418156834672757289427103377, 26.71449342128310790370889286226, 27.19811713004267570547121657862

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