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Properties

Label 1-141-141.140-r0-0-0
Degree $1$
Conductor $141$
Sign $1$
Analytic cond. $0.654801$
Root an. cond. $0.654801$
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 + 4-s + 5-s + 7-s − 8-s − 10-s + 11-s − 13-s − 14-s + 16-s − 17-s − 19-s + 20-s − 22-s + 23-s + 25-s + 26-s + 28-s + 29-s − 31-s − 32-s + 34-s + 35-s + 37-s + 38-s − 40-s + 41-s + ⋯
L(s)  = 1  − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 11-s − 13-s − 14-s + 16-s − 17-s − 19-s + 20-s − 22-s + 23-s + 25-s + 26-s + 28-s + 29-s − 31-s − 32-s + 34-s + 35-s + 37-s + 38-s − 40-s + 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(141\)    =    \(3 \cdot 47\)
Sign: $1$
Analytic conductor: \(0.654801\)
Root analytic conductor: \(0.654801\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{141} (140, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 141,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9146531959\)
\(L(\frac12)\) \(\approx\) \(0.9146531959\)
\(L(1)\) \(\approx\) \(0.8837536145\)
\(L(1)\) \(\approx\) \(0.8837536145\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
47 \( 1 \)
good2 \( 1 - T \)
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 \( 1 - T \)
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

−28.28034701858071653761426580842, −27.27064786791165554305093878545, −26.62845133072383447821058589688, −25.22684137789013209493432149452, −24.85021103675585177317038857414, −23.80533546403154222390393125124, −22.01255854660460357426978712404, −21.31789626582623517015831542051, −20.24088151428339319650670540797, −19.30105743134591090932436474025, −18.02348940921448252470123410333, −17.37053501243346412382136185715, −16.70228881691235905050191818931, −15.03703783429458779475224096795, −14.36496240002823508988875699824, −12.780019605161583539169433407743, −11.48718651241848184314831610907, −10.57634444761654785768117402826, −9.38385653874548965431961361486, −8.613792183503355336165199245632, −7.19064404919162552294694506552, −6.17905267233177715071654023828, −4.69207462835648198131016771377, −2.51835981906200218552874795700, −1.479690742896269333732926672274, 1.479690742896269333732926672274, 2.51835981906200218552874795700, 4.69207462835648198131016771377, 6.17905267233177715071654023828, 7.19064404919162552294694506552, 8.613792183503355336165199245632, 9.38385653874548965431961361486, 10.57634444761654785768117402826, 11.48718651241848184314831610907, 12.780019605161583539169433407743, 14.36496240002823508988875699824, 15.03703783429458779475224096795, 16.70228881691235905050191818931, 17.37053501243346412382136185715, 18.02348940921448252470123410333, 19.30105743134591090932436474025, 20.24088151428339319650670540797, 21.31789626582623517015831542051, 22.01255854660460357426978712404, 23.80533546403154222390393125124, 24.85021103675585177317038857414, 25.22684137789013209493432149452, 26.62845133072383447821058589688, 27.27064786791165554305093878545, 28.28034701858071653761426580842

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