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Properties

Label 1-177-177.176-r0-0-0
Degree $1$
Conductor $177$
Sign $1$
Analytic cond. $0.821984$
Root an. cond. $0.821984$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(0.821984\)
Root analytic conductor: \(0.821984\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.996830388\)
\(L(\frac12)\) \(\approx\) \(1.996830388\)
\(L(1)\) \(\approx\) \(1.764088612\)
\(L(1)\) \(\approx\) \(1.764088612\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
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 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 - T \)
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

−27.31696180732901736560535970026, −26.606110279424871832018465984677, −24.98279352712284464819973114001, −24.365110437664909921841977962622, −23.65836310355462013613294582931, −22.46048096175837297583481149143, −21.939351681002136742391587483161, −20.52054708792274619308272857352, −19.99132304039048574770848572515, −18.93007382476736748537887232246, −17.38998587206040586733580332857, −16.44671827977191895952402199705, −15.164379969689450073200407189101, −14.74137924127001878573599351679, −13.59985188314712592307888746080, −12.25178744141490342655258106558, −11.605440632815519760731482107875, −10.79435581236613206981455667745, −9.00184408557346789537165224322, −7.60217828928878702711526400989, −6.91062554962784503814908170304, −5.22826214290826898120004877159, −4.4057221399634563742025746049, −3.27996355053008719858940199825, −1.68954908042245095697813362495, 1.68954908042245095697813362495, 3.27996355053008719858940199825, 4.4057221399634563742025746049, 5.22826214290826898120004877159, 6.91062554962784503814908170304, 7.60217828928878702711526400989, 9.00184408557346789537165224322, 10.79435581236613206981455667745, 11.605440632815519760731482107875, 12.25178744141490342655258106558, 13.59985188314712592307888746080, 14.74137924127001878573599351679, 15.164379969689450073200407189101, 16.44671827977191895952402199705, 17.38998587206040586733580332857, 18.93007382476736748537887232246, 19.99132304039048574770848572515, 20.52054708792274619308272857352, 21.939351681002136742391587483161, 22.46048096175837297583481149143, 23.65836310355462013613294582931, 24.365110437664909921841977962622, 24.98279352712284464819973114001, 26.606110279424871832018465984677, 27.31696180732901736560535970026

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