Properties

Label 2-177-1.1-c9-0-45
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 26.8·2-s + 81·3-s + 210.·4-s + 1.00e3·5-s + 2.17e3·6-s + 2.54e3·7-s − 8.10e3·8-s + 6.56e3·9-s + 2.70e4·10-s − 1.70e4·11-s + 1.70e4·12-s + 1.77e5·13-s + 6.82e4·14-s + 8.13e4·15-s − 3.25e5·16-s − 4.94e4·17-s + 1.76e5·18-s + 5.60e5·19-s + 2.11e5·20-s + 2.05e5·21-s − 4.58e5·22-s − 6.59e5·23-s − 6.56e5·24-s − 9.44e5·25-s + 4.76e6·26-s + 5.31e5·27-s + 5.35e5·28-s + ⋯
L(s)  = 1  + 1.18·2-s + 0.577·3-s + 0.411·4-s + 0.718·5-s + 0.685·6-s + 0.399·7-s − 0.699·8-s + 0.333·9-s + 0.853·10-s − 0.351·11-s + 0.237·12-s + 1.72·13-s + 0.475·14-s + 0.414·15-s − 1.24·16-s − 0.143·17-s + 0.396·18-s + 0.987·19-s + 0.295·20-s + 0.230·21-s − 0.417·22-s − 0.491·23-s − 0.403·24-s − 0.483·25-s + 2.04·26-s + 0.192·27-s + 0.164·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(6.333788695\)
\(L(\frac12)\) \(\approx\) \(6.333788695\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 81T \)
59 \( 1 + 1.21e7T \)
good2 \( 1 - 26.8T + 512T^{2} \)
5 \( 1 - 1.00e3T + 1.95e6T^{2} \)
7 \( 1 - 2.54e3T + 4.03e7T^{2} \)
11 \( 1 + 1.70e4T + 2.35e9T^{2} \)
13 \( 1 - 1.77e5T + 1.06e10T^{2} \)
17 \( 1 + 4.94e4T + 1.18e11T^{2} \)
19 \( 1 - 5.60e5T + 3.22e11T^{2} \)
23 \( 1 + 6.59e5T + 1.80e12T^{2} \)
29 \( 1 - 4.93e6T + 1.45e13T^{2} \)
31 \( 1 - 7.19e6T + 2.64e13T^{2} \)
37 \( 1 + 8.70e6T + 1.29e14T^{2} \)
41 \( 1 - 3.16e7T + 3.27e14T^{2} \)
43 \( 1 - 7.46e6T + 5.02e14T^{2} \)
47 \( 1 - 9.15e6T + 1.11e15T^{2} \)
53 \( 1 + 2.75e6T + 3.29e15T^{2} \)
61 \( 1 - 1.29e8T + 1.16e16T^{2} \)
67 \( 1 + 2.48e8T + 2.72e16T^{2} \)
71 \( 1 - 2.49e8T + 4.58e16T^{2} \)
73 \( 1 - 1.10e8T + 5.88e16T^{2} \)
79 \( 1 - 6.21e7T + 1.19e17T^{2} \)
83 \( 1 + 1.85e8T + 1.86e17T^{2} \)
89 \( 1 - 3.17e8T + 3.50e17T^{2} \)
97 \( 1 - 2.93e8T + 7.60e17T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.22214660034819007997690343816, −10.01774675369191099171178931244, −8.949492754213263439179528943265, −8.028202861733753515861912320898, −6.45193084644361543314158150446, −5.66038460383325707607575720283, −4.52536595730593628244785151508, −3.48242310796103419342116464684, −2.43271531151244374933583007368, −1.08822027364610436890829653083, 1.08822027364610436890829653083, 2.43271531151244374933583007368, 3.48242310796103419342116464684, 4.52536595730593628244785151508, 5.66038460383325707607575720283, 6.45193084644361543314158150446, 8.028202861733753515861912320898, 8.949492754213263439179528943265, 10.01774675369191099171178931244, 11.22214660034819007997690343816

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