Properties

Label 2-177-1.1-c7-0-1
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  − 20.9·2-s − 27·3-s + 312.·4-s − 465.·5-s + 566.·6-s + 286.·7-s − 3.86e3·8-s + 729·9-s + 9.77e3·10-s − 3.49e3·11-s − 8.43e3·12-s − 4.67e3·13-s − 6.01e3·14-s + 1.25e4·15-s + 4.12e4·16-s − 1.89e3·17-s − 1.52e4·18-s + 6.39e3·19-s − 1.45e5·20-s − 7.73e3·21-s + 7.34e4·22-s + 9.37e3·23-s + 1.04e5·24-s + 1.38e5·25-s + 9.81e4·26-s − 1.96e4·27-s + 8.95e4·28-s + ⋯
L(s)  = 1  − 1.85·2-s − 0.577·3-s + 2.44·4-s − 1.66·5-s + 1.07·6-s + 0.315·7-s − 2.67·8-s + 0.333·9-s + 3.09·10-s − 0.792·11-s − 1.40·12-s − 0.590·13-s − 0.585·14-s + 0.961·15-s + 2.51·16-s − 0.0937·17-s − 0.618·18-s + 0.213·19-s − 4.06·20-s − 0.182·21-s + 1.46·22-s + 0.160·23-s + 1.54·24-s + 1.77·25-s + 1.09·26-s − 0.192·27-s + 0.770·28-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

\(L(4)\) \(\approx\) \(0.01122632701\)
\(L(\frac12)\) \(\approx\) \(0.01122632701\)
\(L(\frac{9}{2})\) 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 \( 1 + 20.9T + 128T^{2} \)
5 \( 1 + 465.T + 7.81e4T^{2} \)
7 \( 1 - 286.T + 8.23e5T^{2} \)
11 \( 1 + 3.49e3T + 1.94e7T^{2} \)
13 \( 1 + 4.67e3T + 6.27e7T^{2} \)
17 \( 1 + 1.89e3T + 4.10e8T^{2} \)
19 \( 1 - 6.39e3T + 8.93e8T^{2} \)
23 \( 1 - 9.37e3T + 3.40e9T^{2} \)
29 \( 1 + 1.60e5T + 1.72e10T^{2} \)
31 \( 1 + 2.72e5T + 2.75e10T^{2} \)
37 \( 1 + 3.53e4T + 9.49e10T^{2} \)
41 \( 1 + 7.27e5T + 1.94e11T^{2} \)
43 \( 1 + 7.12e5T + 2.71e11T^{2} \)
47 \( 1 + 4.76e5T + 5.06e11T^{2} \)
53 \( 1 + 1.49e6T + 1.17e12T^{2} \)
61 \( 1 + 1.04e6T + 3.14e12T^{2} \)
67 \( 1 + 2.56e6T + 6.06e12T^{2} \)
71 \( 1 - 3.16e6T + 9.09e12T^{2} \)
73 \( 1 - 1.62e6T + 1.10e13T^{2} \)
79 \( 1 + 2.11e6T + 1.92e13T^{2} \)
83 \( 1 - 5.43e6T + 2.71e13T^{2} \)
89 \( 1 + 2.27e6T + 4.42e13T^{2} \)
97 \( 1 - 1.00e7T + 8.07e13T^{2} \)
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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

−11.22599231472251240522203901729, −10.46981519554735375365020416810, −9.324716941977065099336520621826, −8.158528559611738195209718124026, −7.61843258698824190857360768868, −6.79801146693778869505102177946, −5.07295695167313938412824764780, −3.33937797332674498339495204430, −1.67486529059361995549623860224, −0.07762229341996068008928851978, 0.07762229341996068008928851978, 1.67486529059361995549623860224, 3.33937797332674498339495204430, 5.07295695167313938412824764780, 6.79801146693778869505102177946, 7.61843258698824190857360768868, 8.158528559611738195209718124026, 9.324716941977065099336520621826, 10.46981519554735375365020416810, 11.22599231472251240522203901729

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