Properties

Label 2-177-1.1-c7-0-13
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  − 16.7·2-s + 27·3-s + 151.·4-s − 298.·5-s − 451.·6-s + 200.·7-s − 395.·8-s + 729·9-s + 4.99e3·10-s + 6.81e3·11-s + 4.09e3·12-s − 7.65e3·13-s − 3.34e3·14-s − 8.06e3·15-s − 1.27e4·16-s − 2.38e4·17-s − 1.21e4·18-s + 4.14e4·19-s − 4.53e4·20-s + 5.40e3·21-s − 1.14e5·22-s + 8.70e4·23-s − 1.06e4·24-s + 1.11e4·25-s + 1.27e5·26-s + 1.96e4·27-s + 3.03e4·28-s + ⋯
L(s)  = 1  − 1.47·2-s + 0.577·3-s + 1.18·4-s − 1.06·5-s − 0.853·6-s + 0.220·7-s − 0.273·8-s + 0.333·9-s + 1.58·10-s + 1.54·11-s + 0.684·12-s − 0.966·13-s − 0.325·14-s − 0.617·15-s − 0.780·16-s − 1.17·17-s − 0.492·18-s + 1.38·19-s − 1.26·20-s + 0.127·21-s − 2.28·22-s + 1.49·23-s − 0.157·24-s + 0.142·25-s + 1.42·26-s + 0.192·27-s + 0.261·28-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

\(L(4)\) \(\approx\) \(0.9017068366\)
\(L(\frac12)\) \(\approx\) \(0.9017068366\)
\(L(\frac{9}{2})\) 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 \( 1 + 16.7T + 128T^{2} \)
5 \( 1 + 298.T + 7.81e4T^{2} \)
7 \( 1 - 200.T + 8.23e5T^{2} \)
11 \( 1 - 6.81e3T + 1.94e7T^{2} \)
13 \( 1 + 7.65e3T + 6.27e7T^{2} \)
17 \( 1 + 2.38e4T + 4.10e8T^{2} \)
19 \( 1 - 4.14e4T + 8.93e8T^{2} \)
23 \( 1 - 8.70e4T + 3.40e9T^{2} \)
29 \( 1 + 2.88e4T + 1.72e10T^{2} \)
31 \( 1 + 1.36e5T + 2.75e10T^{2} \)
37 \( 1 + 2.38e5T + 9.49e10T^{2} \)
41 \( 1 + 3.87e5T + 1.94e11T^{2} \)
43 \( 1 + 5.60e5T + 2.71e11T^{2} \)
47 \( 1 - 3.43e5T + 5.06e11T^{2} \)
53 \( 1 - 1.90e6T + 1.17e12T^{2} \)
61 \( 1 - 2.52e6T + 3.14e12T^{2} \)
67 \( 1 - 1.23e6T + 6.06e12T^{2} \)
71 \( 1 + 5.34e6T + 9.09e12T^{2} \)
73 \( 1 + 4.51e6T + 1.10e13T^{2} \)
79 \( 1 + 1.40e6T + 1.92e13T^{2} \)
83 \( 1 - 8.95e6T + 2.71e13T^{2} \)
89 \( 1 + 1.30e6T + 4.42e13T^{2} \)
97 \( 1 - 1.27e7T + 8.07e13T^{2} \)
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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

−11.39700680333855244614666168574, −10.09177234895831039186764351413, −9.100287635358296208236022878619, −8.618419915262063348325177758172, −7.35488139744909386303438175239, −6.99776156123471334884123338452, −4.71392027730746754608596794301, −3.45591132981231245220935740903, −1.84728409290162336583703382106, −0.64447823028612992828617982427, 0.64447823028612992828617982427, 1.84728409290162336583703382106, 3.45591132981231245220935740903, 4.71392027730746754608596794301, 6.99776156123471334884123338452, 7.35488139744909386303438175239, 8.618419915262063348325177758172, 9.100287635358296208236022878619, 10.09177234895831039186764351413, 11.39700680333855244614666168574

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