Properties

Label 2-177-1.1-c11-0-63
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  + 61.6·2-s + 243·3-s + 1.75e3·4-s + 1.30e4·5-s + 1.49e4·6-s − 4.78e4·7-s − 1.79e4·8-s + 5.90e4·9-s + 8.07e5·10-s + 2.01e5·11-s + 4.26e5·12-s − 1.26e6·13-s − 2.95e6·14-s + 3.18e6·15-s − 4.70e6·16-s + 7.71e6·17-s + 3.64e6·18-s − 1.20e6·19-s + 2.29e7·20-s − 1.16e7·21-s + 1.24e7·22-s + 2.66e7·23-s − 4.37e6·24-s + 1.22e8·25-s − 7.78e7·26-s + 1.43e7·27-s − 8.41e7·28-s + ⋯
L(s)  = 1  + 1.36·2-s + 0.577·3-s + 0.857·4-s + 1.87·5-s + 0.786·6-s − 1.07·7-s − 0.194·8-s + 0.333·9-s + 2.55·10-s + 0.376·11-s + 0.495·12-s − 0.942·13-s − 1.46·14-s + 1.08·15-s − 1.12·16-s + 1.31·17-s + 0.454·18-s − 0.111·19-s + 1.60·20-s − 0.621·21-s + 0.513·22-s + 0.862·23-s − 0.112·24-s + 2.51·25-s − 1.28·26-s + 0.192·27-s − 0.923·28-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

\(L(6)\) \(\approx\) \(8.107578148\)
\(L(\frac12)\) \(\approx\) \(8.107578148\)
\(L(\frac{13}{2})\) 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 \( 1 - 61.6T + 2.04e3T^{2} \)
5 \( 1 - 1.30e4T + 4.88e7T^{2} \)
7 \( 1 + 4.78e4T + 1.97e9T^{2} \)
11 \( 1 - 2.01e5T + 2.85e11T^{2} \)
13 \( 1 + 1.26e6T + 1.79e12T^{2} \)
17 \( 1 - 7.71e6T + 3.42e13T^{2} \)
19 \( 1 + 1.20e6T + 1.16e14T^{2} \)
23 \( 1 - 2.66e7T + 9.52e14T^{2} \)
29 \( 1 - 1.86e8T + 1.22e16T^{2} \)
31 \( 1 + 3.03e7T + 2.54e16T^{2} \)
37 \( 1 - 1.83e8T + 1.77e17T^{2} \)
41 \( 1 - 1.47e9T + 5.50e17T^{2} \)
43 \( 1 - 1.13e9T + 9.29e17T^{2} \)
47 \( 1 - 2.62e9T + 2.47e18T^{2} \)
53 \( 1 - 7.97e8T + 9.26e18T^{2} \)
61 \( 1 + 6.19e9T + 4.35e19T^{2} \)
67 \( 1 + 6.61e9T + 1.22e20T^{2} \)
71 \( 1 + 5.69e9T + 2.31e20T^{2} \)
73 \( 1 + 1.82e10T + 3.13e20T^{2} \)
79 \( 1 + 1.19e10T + 7.47e20T^{2} \)
83 \( 1 - 7.91e9T + 1.28e21T^{2} \)
89 \( 1 + 8.66e10T + 2.77e21T^{2} \)
97 \( 1 + 6.46e10T + 7.15e21T^{2} \)
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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

−10.49005973312165782748007263917, −9.596641496829754822043854657401, −9.075075901414991816742174349911, −7.14585469317355852773895884842, −6.17354822113213951137222379167, −5.53280754554331193246705745617, −4.37004709104652421870109839064, −2.91171361724548971891575160777, −2.58804316578936703453698259461, −1.07073110334885242444656934414, 1.07073110334885242444656934414, 2.58804316578936703453698259461, 2.91171361724548971891575160777, 4.37004709104652421870109839064, 5.53280754554331193246705745617, 6.17354822113213951137222379167, 7.14585469317355852773895884842, 9.075075901414991816742174349911, 9.596641496829754822043854657401, 10.49005973312165782748007263917

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