Properties

Label 2-177-1.1-c7-0-2
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  − 12.7·2-s + 27·3-s + 35.7·4-s − 405.·5-s − 345.·6-s − 375.·7-s + 1.18e3·8-s + 729·9-s + 5.18e3·10-s − 1.86e3·11-s + 966.·12-s + 143.·13-s + 4.80e3·14-s − 1.09e4·15-s − 1.96e4·16-s − 7.99e3·17-s − 9.32e3·18-s − 5.49e4·19-s − 1.45e4·20-s − 1.01e4·21-s + 2.39e4·22-s − 5.72e4·23-s + 3.18e4·24-s + 8.60e4·25-s − 1.84e3·26-s + 1.96e4·27-s − 1.34e4·28-s + ⋯
L(s)  = 1  − 1.13·2-s + 0.577·3-s + 0.279·4-s − 1.44·5-s − 0.653·6-s − 0.414·7-s + 0.814·8-s + 0.333·9-s + 1.63·10-s − 0.423·11-s + 0.161·12-s + 0.0181·13-s + 0.468·14-s − 0.837·15-s − 1.20·16-s − 0.394·17-s − 0.377·18-s − 1.83·19-s − 0.405·20-s − 0.239·21-s + 0.478·22-s − 0.980·23-s + 0.470·24-s + 1.10·25-s − 0.0205·26-s + 0.192·27-s − 0.115·28-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

\(L(4)\) \(\approx\) \(0.2915892976\)
\(L(\frac12)\) \(\approx\) \(0.2915892976\)
\(L(\frac{9}{2})\) 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 \( 1 + 12.7T + 128T^{2} \)
5 \( 1 + 405.T + 7.81e4T^{2} \)
7 \( 1 + 375.T + 8.23e5T^{2} \)
11 \( 1 + 1.86e3T + 1.94e7T^{2} \)
13 \( 1 - 143.T + 6.27e7T^{2} \)
17 \( 1 + 7.99e3T + 4.10e8T^{2} \)
19 \( 1 + 5.49e4T + 8.93e8T^{2} \)
23 \( 1 + 5.72e4T + 3.40e9T^{2} \)
29 \( 1 - 4.41e4T + 1.72e10T^{2} \)
31 \( 1 + 1.27e5T + 2.75e10T^{2} \)
37 \( 1 + 3.43e5T + 9.49e10T^{2} \)
41 \( 1 - 5.73e4T + 1.94e11T^{2} \)
43 \( 1 - 1.31e5T + 2.71e11T^{2} \)
47 \( 1 - 1.33e4T + 5.06e11T^{2} \)
53 \( 1 + 6.42e5T + 1.17e12T^{2} \)
61 \( 1 + 1.24e6T + 3.14e12T^{2} \)
67 \( 1 - 3.33e5T + 6.06e12T^{2} \)
71 \( 1 - 4.40e6T + 9.09e12T^{2} \)
73 \( 1 + 7.90e5T + 1.10e13T^{2} \)
79 \( 1 - 1.80e6T + 1.92e13T^{2} \)
83 \( 1 - 3.39e6T + 2.71e13T^{2} \)
89 \( 1 - 9.39e6T + 4.42e13T^{2} \)
97 \( 1 - 7.73e5T + 8.07e13T^{2} \)
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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

−11.04678114705309396518403025269, −10.32171139956573210416399570965, −9.134208856738786056981161668152, −8.323310379771383318996487123235, −7.74028813864141995767502578242, −6.66134133958015655981804426446, −4.57275394903221384920408073174, −3.64744733633662512931180877205, −2.04371325526745881959284738884, −0.32814966425222134135344191912, 0.32814966425222134135344191912, 2.04371325526745881959284738884, 3.64744733633662512931180877205, 4.57275394903221384920408073174, 6.66134133958015655981804426446, 7.74028813864141995767502578242, 8.323310379771383318996487123235, 9.134208856738786056981161668152, 10.32171139956573210416399570965, 11.04678114705309396518403025269

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