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Properties

Label 1-193-193.192-r0-0-0
Degree $1$
Conductor $193$
Sign $1$
Analytic cond. $0.896288$
Root an. cond. $0.896288$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 11-s + 12-s − 13-s + 14-s − 15-s + 16-s − 17-s + 18-s − 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s − 26-s + 27-s + 28-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 11-s + 12-s − 13-s + 14-s − 15-s + 16-s − 17-s + 18-s − 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s − 26-s + 27-s + 28-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $1$
Analytic conductor: \(0.896288\)
Root analytic conductor: \(0.896288\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{193} (192, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.552318543\)
\(L(\frac12)\) \(\approx\) \(2.552318543\)
\(L(1)\) \(\approx\) \(2.170434019\)
\(L(1)\) \(\approx\) \(2.170434019\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad193 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 + T \)
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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

−26.85428018974484244472829809726, −26.09316207558873819966435212144, −24.76823073400759295034747600107, −24.20090178904136891323353487456, −23.471606309764255660425986199256, −22.251835267191131263040604190783, −21.11985901577472344332870388753, −20.584586033230579068939114532677, −19.591702178345991021223476457188, −18.827534843073612204822428583289, −17.23418506093523534080217468230, −15.79384921384217512612051863610, −15.131490182507879520693448637936, −14.54362432235762068985577321694, −13.32535990200728667229500907268, −12.511924449852632123882858543211, −11.3326439812807972770548893929, −10.42134772259636973423360584901, −8.6180956568955563766931993336, −7.73672835482298048291918630126, −6.95454016967741322889029040878, −4.96143288246178556517209049509, −4.34123204697491882342583335037, −3.01285271901108115357928066407, −1.99262676052092786030892794973, 1.99262676052092786030892794973, 3.01285271901108115357928066407, 4.34123204697491882342583335037, 4.96143288246178556517209049509, 6.95454016967741322889029040878, 7.73672835482298048291918630126, 8.6180956568955563766931993336, 10.42134772259636973423360584901, 11.3326439812807972770548893929, 12.511924449852632123882858543211, 13.32535990200728667229500907268, 14.54362432235762068985577321694, 15.131490182507879520693448637936, 15.79384921384217512612051863610, 17.23418506093523534080217468230, 18.827534843073612204822428583289, 19.591702178345991021223476457188, 20.584586033230579068939114532677, 21.11985901577472344332870388753, 22.251835267191131263040604190783, 23.471606309764255660425986199256, 24.20090178904136891323353487456, 24.76823073400759295034747600107, 26.09316207558873819966435212144, 26.85428018974484244472829809726

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