Properties

Label 2-177-1.1-c11-0-96
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 60.5·2-s − 243·3-s + 1.62e3·4-s + 3.73e3·5-s − 1.47e4·6-s + 5.15e4·7-s − 2.58e4·8-s + 5.90e4·9-s + 2.26e5·10-s + 8.19e3·11-s − 3.93e5·12-s − 2.10e6·13-s + 3.12e6·14-s − 9.07e5·15-s − 4.88e6·16-s + 4.38e6·17-s + 3.57e6·18-s − 3.78e6·19-s + 6.05e6·20-s − 1.25e7·21-s + 4.96e5·22-s + 2.08e7·23-s + 6.28e6·24-s − 3.48e7·25-s − 1.27e8·26-s − 1.43e7·27-s + 8.35e7·28-s + ⋯
L(s)  = 1  + 1.33·2-s − 0.577·3-s + 0.791·4-s + 0.534·5-s − 0.772·6-s + 1.15·7-s − 0.279·8-s + 0.333·9-s + 0.715·10-s + 0.0153·11-s − 0.456·12-s − 1.56·13-s + 1.55·14-s − 0.308·15-s − 1.16·16-s + 0.749·17-s + 0.446·18-s − 0.351·19-s + 0.422·20-s − 0.669·21-s + 0.0205·22-s + 0.676·23-s + 0.161·24-s − 0.714·25-s − 2.10·26-s − 0.192·27-s + 0.917·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 243T \)
59 \( 1 - 7.14e8T \)
good2 \( 1 - 60.5T + 2.04e3T^{2} \)
5 \( 1 - 3.73e3T + 4.88e7T^{2} \)
7 \( 1 - 5.15e4T + 1.97e9T^{2} \)
11 \( 1 - 8.19e3T + 2.85e11T^{2} \)
13 \( 1 + 2.10e6T + 1.79e12T^{2} \)
17 \( 1 - 4.38e6T + 3.42e13T^{2} \)
19 \( 1 + 3.78e6T + 1.16e14T^{2} \)
23 \( 1 - 2.08e7T + 9.52e14T^{2} \)
29 \( 1 + 1.72e8T + 1.22e16T^{2} \)
31 \( 1 - 2.05e8T + 2.54e16T^{2} \)
37 \( 1 + 4.55e7T + 1.77e17T^{2} \)
41 \( 1 + 1.14e9T + 5.50e17T^{2} \)
43 \( 1 - 6.27e8T + 9.29e17T^{2} \)
47 \( 1 + 1.39e9T + 2.47e18T^{2} \)
53 \( 1 - 5.76e9T + 9.26e18T^{2} \)
61 \( 1 + 9.92e9T + 4.35e19T^{2} \)
67 \( 1 - 2.96e9T + 1.22e20T^{2} \)
71 \( 1 + 3.75e9T + 2.31e20T^{2} \)
73 \( 1 + 1.30e10T + 3.13e20T^{2} \)
79 \( 1 - 1.07e10T + 7.47e20T^{2} \)
83 \( 1 - 3.38e10T + 1.28e21T^{2} \)
89 \( 1 + 7.13e10T + 2.77e21T^{2} \)
97 \( 1 + 5.74e10T + 7.15e21T^{2} \)
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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

−10.39785737248621836005615770562, −9.340402814772553471699365529999, −7.85585320620936247781722286938, −6.75030450999553987419907841532, −5.51113434880213799927948839913, −5.07124916326801583088046527422, −4.10593209374431261898965247147, −2.65436443240584474947205936577, −1.59578672301398120187227618343, 0, 1.59578672301398120187227618343, 2.65436443240584474947205936577, 4.10593209374431261898965247147, 5.07124916326801583088046527422, 5.51113434880213799927948839913, 6.75030450999553987419907841532, 7.85585320620936247781722286938, 9.340402814772553471699365529999, 10.39785737248621836005615770562

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