Properties

Label 2-177-1.1-c7-0-14
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $55.2921$
Root an. cond. $7.43586$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.2·2-s + 27·3-s − 23.7·4-s − 397.·5-s + 275.·6-s − 505.·7-s − 1.54e3·8-s + 729·9-s − 4.05e3·10-s − 5.49e3·11-s − 642.·12-s + 5.26e3·13-s − 5.16e3·14-s − 1.07e4·15-s − 1.27e4·16-s + 3.36e4·17-s + 7.44e3·18-s + 4.18e4·19-s + 9.46e3·20-s − 1.36e4·21-s − 5.60e4·22-s − 1.47e4·23-s − 4.18e4·24-s + 8.00e4·25-s + 5.37e4·26-s + 1.96e4·27-s + 1.20e4·28-s + ⋯
L(s)  = 1  + 0.902·2-s + 0.577·3-s − 0.185·4-s − 1.42·5-s + 0.520·6-s − 0.557·7-s − 1.07·8-s + 0.333·9-s − 1.28·10-s − 1.24·11-s − 0.107·12-s + 0.664·13-s − 0.502·14-s − 0.821·15-s − 0.779·16-s + 1.66·17-s + 0.300·18-s + 1.39·19-s + 0.264·20-s − 0.321·21-s − 1.12·22-s − 0.252·23-s − 0.617·24-s + 1.02·25-s + 0.599·26-s + 0.192·27-s + 0.103·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(55.2921\)
Root analytic conductor: \(7.43586\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(2.067525245\)
\(L(\frac12)\) \(\approx\) \(2.067525245\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 27T \)
59 \( 1 - 2.05e5T \)
good2 \( 1 - 10.2T + 128T^{2} \)
5 \( 1 + 397.T + 7.81e4T^{2} \)
7 \( 1 + 505.T + 8.23e5T^{2} \)
11 \( 1 + 5.49e3T + 1.94e7T^{2} \)
13 \( 1 - 5.26e3T + 6.27e7T^{2} \)
17 \( 1 - 3.36e4T + 4.10e8T^{2} \)
19 \( 1 - 4.18e4T + 8.93e8T^{2} \)
23 \( 1 + 1.47e4T + 3.40e9T^{2} \)
29 \( 1 - 1.92e5T + 1.72e10T^{2} \)
31 \( 1 + 2.22e5T + 2.75e10T^{2} \)
37 \( 1 - 3.19e5T + 9.49e10T^{2} \)
41 \( 1 + 2.10e5T + 1.94e11T^{2} \)
43 \( 1 + 1.97e5T + 2.71e11T^{2} \)
47 \( 1 - 1.24e6T + 5.06e11T^{2} \)
53 \( 1 + 3.46e5T + 1.17e12T^{2} \)
61 \( 1 + 2.78e6T + 3.14e12T^{2} \)
67 \( 1 + 4.61e6T + 6.06e12T^{2} \)
71 \( 1 - 1.49e6T + 9.09e12T^{2} \)
73 \( 1 + 1.38e6T + 1.10e13T^{2} \)
79 \( 1 - 7.13e6T + 1.92e13T^{2} \)
83 \( 1 - 2.03e6T + 2.71e13T^{2} \)
89 \( 1 - 5.52e6T + 4.42e13T^{2} \)
97 \( 1 - 1.50e7T + 8.07e13T^{2} \)
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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

−11.82604023333759167043368010968, −10.44752245594387757352348214344, −9.300610833843144949506683226228, −8.112712198793032521076140097378, −7.47509579047455931165323794589, −5.83673531977203186419235700212, −4.71860929748580653526884039620, −3.48902821247304872028428202814, −3.08696547030317116808625043204, −0.66240305464672685674841881425, 0.66240305464672685674841881425, 3.08696547030317116808625043204, 3.48902821247304872028428202814, 4.71860929748580653526884039620, 5.83673531977203186419235700212, 7.47509579047455931165323794589, 8.112712198793032521076140097378, 9.300610833843144949506683226228, 10.44752245594387757352348214344, 11.82604023333759167043368010968

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