Properties

Label 2-177-1.1-c7-0-19
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  − 5.66·2-s + 27·3-s − 95.8·4-s − 15.0·5-s − 152.·6-s + 499.·7-s + 1.26e3·8-s + 729·9-s + 85.0·10-s + 5.25e3·11-s − 2.58e3·12-s + 114.·13-s − 2.82e3·14-s − 405.·15-s + 5.08e3·16-s − 1.67e4·17-s − 4.13e3·18-s + 5.40e3·19-s + 1.43e3·20-s + 1.34e4·21-s − 2.97e4·22-s − 7.46e4·23-s + 3.42e4·24-s − 7.78e4·25-s − 649.·26-s + 1.96e4·27-s − 4.78e4·28-s + ⋯
L(s)  = 1  − 0.500·2-s + 0.577·3-s − 0.749·4-s − 0.0537·5-s − 0.289·6-s + 0.550·7-s + 0.876·8-s + 0.333·9-s + 0.0269·10-s + 1.18·11-s − 0.432·12-s + 0.0144·13-s − 0.275·14-s − 0.0310·15-s + 0.310·16-s − 0.827·17-s − 0.166·18-s + 0.180·19-s + 0.0402·20-s + 0.317·21-s − 0.595·22-s − 1.27·23-s + 0.505·24-s − 0.997·25-s − 0.00724·26-s + 0.192·27-s − 0.412·28-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

\(L(4)\) \(\approx\) \(1.799763126\)
\(L(\frac12)\) \(\approx\) \(1.799763126\)
\(L(\frac{9}{2})\) 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 \( 1 + 5.66T + 128T^{2} \)
5 \( 1 + 15.0T + 7.81e4T^{2} \)
7 \( 1 - 499.T + 8.23e5T^{2} \)
11 \( 1 - 5.25e3T + 1.94e7T^{2} \)
13 \( 1 - 114.T + 6.27e7T^{2} \)
17 \( 1 + 1.67e4T + 4.10e8T^{2} \)
19 \( 1 - 5.40e3T + 8.93e8T^{2} \)
23 \( 1 + 7.46e4T + 3.40e9T^{2} \)
29 \( 1 - 5.87e4T + 1.72e10T^{2} \)
31 \( 1 - 1.94e5T + 2.75e10T^{2} \)
37 \( 1 - 3.19e5T + 9.49e10T^{2} \)
41 \( 1 - 1.72e5T + 1.94e11T^{2} \)
43 \( 1 - 1.74e4T + 2.71e11T^{2} \)
47 \( 1 - 7.01e5T + 5.06e11T^{2} \)
53 \( 1 - 4.67e5T + 1.17e12T^{2} \)
61 \( 1 + 8.78e5T + 3.14e12T^{2} \)
67 \( 1 + 3.01e5T + 6.06e12T^{2} \)
71 \( 1 - 7.22e5T + 9.09e12T^{2} \)
73 \( 1 - 6.53e6T + 1.10e13T^{2} \)
79 \( 1 - 3.78e6T + 1.92e13T^{2} \)
83 \( 1 + 1.90e5T + 2.71e13T^{2} \)
89 \( 1 - 2.70e6T + 4.42e13T^{2} \)
97 \( 1 - 1.47e7T + 8.07e13T^{2} \)
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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

−11.33090988579566056503642348559, −10.06942354045674495816755777290, −9.297819797052612609110765575251, −8.426163028828170731718469649708, −7.65617356459037656778195511058, −6.24209770087565557310235589969, −4.61863700294320884520812528100, −3.84274701158017548411731992385, −2.03200818194783062775943970416, −0.807665615614656166019043310415, 0.807665615614656166019043310415, 2.03200818194783062775943970416, 3.84274701158017548411731992385, 4.61863700294320884520812528100, 6.24209770087565557310235589969, 7.65617356459037656778195511058, 8.426163028828170731718469649708, 9.297819797052612609110765575251, 10.06942354045674495816755777290, 11.33090988579566056503642348559

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