Properties

Label 2-177-1.1-c5-0-13
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.3·2-s + 9·3-s + 75.5·4-s + 33.2·5-s − 93.3·6-s − 56.9·7-s − 451.·8-s + 81·9-s − 344.·10-s + 432.·11-s + 680.·12-s + 178.·13-s + 590.·14-s + 299.·15-s + 2.26e3·16-s + 1.65e3·17-s − 840.·18-s − 168.·19-s + 2.51e3·20-s − 512.·21-s − 4.48e3·22-s + 143.·23-s − 4.06e3·24-s − 2.01e3·25-s − 1.84e3·26-s + 729·27-s − 4.30e3·28-s + ⋯
L(s)  = 1  − 1.83·2-s + 0.577·3-s + 2.36·4-s + 0.594·5-s − 1.05·6-s − 0.439·7-s − 2.49·8-s + 0.333·9-s − 1.09·10-s + 1.07·11-s + 1.36·12-s + 0.292·13-s + 0.805·14-s + 0.343·15-s + 2.21·16-s + 1.39·17-s − 0.611·18-s − 0.107·19-s + 1.40·20-s − 0.253·21-s − 1.97·22-s + 0.0564·23-s − 1.44·24-s − 0.646·25-s − 0.535·26-s + 0.192·27-s − 1.03·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.229535091\)
\(L(\frac12)\) \(\approx\) \(1.229535091\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9T \)
59 \( 1 + 3.48e3T \)
good2 \( 1 + 10.3T + 32T^{2} \)
5 \( 1 - 33.2T + 3.12e3T^{2} \)
7 \( 1 + 56.9T + 1.68e4T^{2} \)
11 \( 1 - 432.T + 1.61e5T^{2} \)
13 \( 1 - 178.T + 3.71e5T^{2} \)
17 \( 1 - 1.65e3T + 1.41e6T^{2} \)
19 \( 1 + 168.T + 2.47e6T^{2} \)
23 \( 1 - 143.T + 6.43e6T^{2} \)
29 \( 1 - 275.T + 2.05e7T^{2} \)
31 \( 1 + 2.27e3T + 2.86e7T^{2} \)
37 \( 1 + 1.12e3T + 6.93e7T^{2} \)
41 \( 1 + 2.11e3T + 1.15e8T^{2} \)
43 \( 1 - 3.84e3T + 1.47e8T^{2} \)
47 \( 1 - 1.48e4T + 2.29e8T^{2} \)
53 \( 1 - 9.65e3T + 4.18e8T^{2} \)
61 \( 1 - 2.92e4T + 8.44e8T^{2} \)
67 \( 1 - 1.50e4T + 1.35e9T^{2} \)
71 \( 1 - 1.32e4T + 1.80e9T^{2} \)
73 \( 1 - 4.24e4T + 2.07e9T^{2} \)
79 \( 1 + 5.55e4T + 3.07e9T^{2} \)
83 \( 1 - 8.10e4T + 3.93e9T^{2} \)
89 \( 1 - 1.07e5T + 5.58e9T^{2} \)
97 \( 1 + 9.24e4T + 8.58e9T^{2} \)
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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

−11.48119377947168429934401953828, −10.28996013320686687164480522584, −9.627575623768760848129252127585, −8.956022896720903408890873363016, −7.954504056934920708469417046334, −6.94088846164375724928511798380, −5.93285946839259268775284678545, −3.47441571646836258728613302216, −2.03692070308396394438693293033, −0.936215857751623622924571040945, 0.936215857751623622924571040945, 2.03692070308396394438693293033, 3.47441571646836258728613302216, 5.93285946839259268775284678545, 6.94088846164375724928511798380, 7.954504056934920708469417046334, 8.956022896720903408890873363016, 9.627575623768760848129252127585, 10.28996013320686687164480522584, 11.48119377947168429934401953828

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