Properties

Label 2-177-1.1-c11-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11.8·2-s − 243·3-s − 1.90e3·4-s − 8.83e3·5-s + 2.87e3·6-s + 1.92e4·7-s + 4.67e4·8-s + 5.90e4·9-s + 1.04e5·10-s + 4.13e5·11-s + 4.63e5·12-s − 2.15e6·13-s − 2.27e5·14-s + 2.14e6·15-s + 3.35e6·16-s − 6.59e6·17-s − 6.97e5·18-s + 7.72e5·19-s + 1.68e7·20-s − 4.66e6·21-s − 4.88e6·22-s + 4.19e7·23-s − 1.13e7·24-s + 2.91e7·25-s + 2.54e7·26-s − 1.43e7·27-s − 3.66e7·28-s + ⋯
L(s)  = 1  − 0.261·2-s − 0.577·3-s − 0.931·4-s − 1.26·5-s + 0.150·6-s + 0.432·7-s + 0.504·8-s + 0.333·9-s + 0.330·10-s + 0.774·11-s + 0.537·12-s − 1.61·13-s − 0.112·14-s + 0.729·15-s + 0.799·16-s − 1.12·17-s − 0.0870·18-s + 0.0715·19-s + 1.17·20-s − 0.249·21-s − 0.202·22-s + 1.36·23-s − 0.291·24-s + 0.597·25-s + 0.420·26-s − 0.192·27-s − 0.402·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.1365801209\)
\(L(\frac12)\) \(\approx\) \(0.1365801209\)
\(L(\frac{13}{2})\) 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 \( 1 + 11.8T + 2.04e3T^{2} \)
5 \( 1 + 8.83e3T + 4.88e7T^{2} \)
7 \( 1 - 1.92e4T + 1.97e9T^{2} \)
11 \( 1 - 4.13e5T + 2.85e11T^{2} \)
13 \( 1 + 2.15e6T + 1.79e12T^{2} \)
17 \( 1 + 6.59e6T + 3.42e13T^{2} \)
19 \( 1 - 7.72e5T + 1.16e14T^{2} \)
23 \( 1 - 4.19e7T + 9.52e14T^{2} \)
29 \( 1 + 4.25e7T + 1.22e16T^{2} \)
31 \( 1 + 6.05e7T + 2.54e16T^{2} \)
37 \( 1 + 5.85e8T + 1.77e17T^{2} \)
41 \( 1 + 2.62e8T + 5.50e17T^{2} \)
43 \( 1 + 1.36e9T + 9.29e17T^{2} \)
47 \( 1 + 9.55e8T + 2.47e18T^{2} \)
53 \( 1 + 3.76e9T + 9.26e18T^{2} \)
61 \( 1 + 1.01e9T + 4.35e19T^{2} \)
67 \( 1 + 1.22e10T + 1.22e20T^{2} \)
71 \( 1 + 2.13e10T + 2.31e20T^{2} \)
73 \( 1 - 7.12e8T + 3.13e20T^{2} \)
79 \( 1 - 4.14e10T + 7.47e20T^{2} \)
83 \( 1 + 4.23e9T + 1.28e21T^{2} \)
89 \( 1 - 7.44e9T + 2.77e21T^{2} \)
97 \( 1 - 1.19e11T + 7.15e21T^{2} \)
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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

−10.76535009153227250227511673974, −9.567160658232733396204227266854, −8.682932920830970883464319669642, −7.64061788086121375085124259943, −6.82107603214799176816674361300, −5.01956499416946896232007328060, −4.56584732858035554907066900179, −3.42407506124500055508934703739, −1.55813441110141297170568092562, −0.18666861834323347580886722680, 0.18666861834323347580886722680, 1.55813441110141297170568092562, 3.42407506124500055508934703739, 4.56584732858035554907066900179, 5.01956499416946896232007328060, 6.82107603214799176816674361300, 7.64061788086121375085124259943, 8.682932920830970883464319669642, 9.567160658232733396204227266854, 10.76535009153227250227511673974

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