Properties

Label 2-177-1.1-c7-0-12
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  − 12.3·2-s − 27·3-s + 24.9·4-s − 477.·5-s + 333.·6-s + 1.34e3·7-s + 1.27e3·8-s + 729·9-s + 5.91e3·10-s + 6.22e3·11-s − 674.·12-s + 1.09e4·13-s − 1.66e4·14-s + 1.29e4·15-s − 1.89e4·16-s − 3.18e4·17-s − 9.01e3·18-s − 3.64e4·19-s − 1.19e4·20-s − 3.63e4·21-s − 7.69e4·22-s + 3.83e4·23-s − 3.44e4·24-s + 1.50e5·25-s − 1.35e5·26-s − 1.96e4·27-s + 3.35e4·28-s + ⋯
L(s)  = 1  − 1.09·2-s − 0.577·3-s + 0.195·4-s − 1.71·5-s + 0.631·6-s + 1.48·7-s + 0.879·8-s + 0.333·9-s + 1.86·10-s + 1.40·11-s − 0.112·12-s + 1.37·13-s − 1.61·14-s + 0.987·15-s − 1.15·16-s − 1.57·17-s − 0.364·18-s − 1.22·19-s − 0.333·20-s − 0.855·21-s − 1.54·22-s + 0.656·23-s − 0.508·24-s + 1.92·25-s − 1.50·26-s − 0.192·27-s + 0.288·28-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

\(L(4)\) \(\approx\) \(0.7145607435\)
\(L(\frac12)\) \(\approx\) \(0.7145607435\)
\(L(\frac{9}{2})\) 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 \( 1 + 12.3T + 128T^{2} \)
5 \( 1 + 477.T + 7.81e4T^{2} \)
7 \( 1 - 1.34e3T + 8.23e5T^{2} \)
11 \( 1 - 6.22e3T + 1.94e7T^{2} \)
13 \( 1 - 1.09e4T + 6.27e7T^{2} \)
17 \( 1 + 3.18e4T + 4.10e8T^{2} \)
19 \( 1 + 3.64e4T + 8.93e8T^{2} \)
23 \( 1 - 3.83e4T + 3.40e9T^{2} \)
29 \( 1 + 1.38e5T + 1.72e10T^{2} \)
31 \( 1 - 2.56e5T + 2.75e10T^{2} \)
37 \( 1 - 2.21e4T + 9.49e10T^{2} \)
41 \( 1 - 3.16e5T + 1.94e11T^{2} \)
43 \( 1 - 1.48e5T + 2.71e11T^{2} \)
47 \( 1 - 7.48e5T + 5.06e11T^{2} \)
53 \( 1 + 1.94e6T + 1.17e12T^{2} \)
61 \( 1 - 1.76e6T + 3.14e12T^{2} \)
67 \( 1 + 4.25e5T + 6.06e12T^{2} \)
71 \( 1 + 3.19e6T + 9.09e12T^{2} \)
73 \( 1 + 4.35e6T + 1.10e13T^{2} \)
79 \( 1 + 2.91e6T + 1.92e13T^{2} \)
83 \( 1 + 1.97e6T + 2.71e13T^{2} \)
89 \( 1 - 7.72e6T + 4.42e13T^{2} \)
97 \( 1 - 9.26e6T + 8.07e13T^{2} \)
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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

−11.12903279020235704545258744620, −10.86013961046493981952669651106, −8.913629382054586826094689804480, −8.516322489286252739619588473442, −7.56408757691146947629541238173, −6.51461793318967654554976716003, −4.45202705040592559313436993850, −4.17077177032802533141134824435, −1.55866135331196834498945166944, −0.61977552223271549743630254131, 0.61977552223271549743630254131, 1.55866135331196834498945166944, 4.17077177032802533141134824435, 4.45202705040592559313436993850, 6.51461793318967654554976716003, 7.56408757691146947629541238173, 8.516322489286252739619588473442, 8.913629382054586826094689804480, 10.86013961046493981952669651106, 11.12903279020235704545258744620

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