Properties

Label 2-177-1.1-c7-0-9
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  − 19.5·2-s + 27·3-s + 253.·4-s − 49.6·5-s − 527.·6-s − 119.·7-s − 2.44e3·8-s + 729·9-s + 968.·10-s − 5.49e3·11-s + 6.83e3·12-s − 1.28e4·13-s + 2.34e3·14-s − 1.33e3·15-s + 1.52e4·16-s + 2.56e4·17-s − 1.42e4·18-s − 1.75e4·19-s − 1.25e4·20-s − 3.23e3·21-s + 1.07e5·22-s + 6.32e4·23-s − 6.59e4·24-s − 7.56e4·25-s + 2.51e5·26-s + 1.96e4·27-s − 3.03e4·28-s + ⋯
L(s)  = 1  − 1.72·2-s + 0.577·3-s + 1.97·4-s − 0.177·5-s − 0.996·6-s − 0.132·7-s − 1.68·8-s + 0.333·9-s + 0.306·10-s − 1.24·11-s + 1.14·12-s − 1.62·13-s + 0.227·14-s − 0.102·15-s + 0.931·16-s + 1.26·17-s − 0.575·18-s − 0.587·19-s − 0.350·20-s − 0.0762·21-s + 2.14·22-s + 1.08·23-s − 0.973·24-s − 0.968·25-s + 2.80·26-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

\(L(4)\) \(\approx\) \(0.6630506810\)
\(L(\frac12)\) \(\approx\) \(0.6630506810\)
\(L(\frac{9}{2})\) 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 \( 1 + 19.5T + 128T^{2} \)
5 \( 1 + 49.6T + 7.81e4T^{2} \)
7 \( 1 + 119.T + 8.23e5T^{2} \)
11 \( 1 + 5.49e3T + 1.94e7T^{2} \)
13 \( 1 + 1.28e4T + 6.27e7T^{2} \)
17 \( 1 - 2.56e4T + 4.10e8T^{2} \)
19 \( 1 + 1.75e4T + 8.93e8T^{2} \)
23 \( 1 - 6.32e4T + 3.40e9T^{2} \)
29 \( 1 + 2.96e4T + 1.72e10T^{2} \)
31 \( 1 - 1.66e5T + 2.75e10T^{2} \)
37 \( 1 - 1.07e5T + 9.49e10T^{2} \)
41 \( 1 - 5.20e5T + 1.94e11T^{2} \)
43 \( 1 + 9.43e5T + 2.71e11T^{2} \)
47 \( 1 + 7.89e5T + 5.06e11T^{2} \)
53 \( 1 + 9.30e5T + 1.17e12T^{2} \)
61 \( 1 - 2.19e6T + 3.14e12T^{2} \)
67 \( 1 - 1.13e6T + 6.06e12T^{2} \)
71 \( 1 - 2.46e6T + 9.09e12T^{2} \)
73 \( 1 - 5.09e6T + 1.10e13T^{2} \)
79 \( 1 - 4.40e6T + 1.92e13T^{2} \)
83 \( 1 - 1.72e5T + 2.71e13T^{2} \)
89 \( 1 - 8.28e6T + 4.42e13T^{2} \)
97 \( 1 + 6.60e6T + 8.07e13T^{2} \)
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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

−10.98894097791555166922934516778, −9.909336334353845320083530499507, −9.617352509951306803225895647875, −8.136298542800830427881435949169, −7.81788429276437906271323538838, −6.75378011417380902995178015020, −5.03327575378250668538539916851, −3.01469621812905367803866553668, −2.05319025205459750964437317412, −0.53852939831388295648646332943, 0.53852939831388295648646332943, 2.05319025205459750964437317412, 3.01469621812905367803866553668, 5.03327575378250668538539916851, 6.75378011417380902995178015020, 7.81788429276437906271323538838, 8.136298542800830427881435949169, 9.617352509951306803225895647875, 9.909336334353845320083530499507, 10.98894097791555166922934516778

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