Properties

Label 2-177-1.1-c7-0-57
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  + 21.8·2-s + 27·3-s + 350.·4-s + 336.·5-s + 590.·6-s − 653.·7-s + 4.87e3·8-s + 729·9-s + 7.35e3·10-s + 2.82e3·11-s + 9.46e3·12-s − 5.70e3·13-s − 1.43e4·14-s + 9.08e3·15-s + 6.17e4·16-s + 2.74e4·17-s + 1.59e4·18-s − 5.26e4·19-s + 1.17e5·20-s − 1.76e4·21-s + 6.18e4·22-s − 1.12e5·23-s + 1.31e5·24-s + 3.49e4·25-s − 1.24e5·26-s + 1.96e4·27-s − 2.29e5·28-s + ⋯
L(s)  = 1  + 1.93·2-s + 0.577·3-s + 2.73·4-s + 1.20·5-s + 1.11·6-s − 0.720·7-s + 3.36·8-s + 0.333·9-s + 2.32·10-s + 0.639·11-s + 1.58·12-s − 0.720·13-s − 1.39·14-s + 0.694·15-s + 3.76·16-s + 1.35·17-s + 0.644·18-s − 1.76·19-s + 3.29·20-s − 0.415·21-s + 1.23·22-s − 1.93·23-s + 1.94·24-s + 0.447·25-s − 1.39·26-s + 0.192·27-s − 1.97·28-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

\(L(4)\) \(\approx\) \(10.85285797\)
\(L(\frac12)\) \(\approx\) \(10.85285797\)
\(L(\frac{9}{2})\) 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 \( 1 - 21.8T + 128T^{2} \)
5 \( 1 - 336.T + 7.81e4T^{2} \)
7 \( 1 + 653.T + 8.23e5T^{2} \)
11 \( 1 - 2.82e3T + 1.94e7T^{2} \)
13 \( 1 + 5.70e3T + 6.27e7T^{2} \)
17 \( 1 - 2.74e4T + 4.10e8T^{2} \)
19 \( 1 + 5.26e4T + 8.93e8T^{2} \)
23 \( 1 + 1.12e5T + 3.40e9T^{2} \)
29 \( 1 - 1.29e5T + 1.72e10T^{2} \)
31 \( 1 - 2.64e5T + 2.75e10T^{2} \)
37 \( 1 + 1.62e5T + 9.49e10T^{2} \)
41 \( 1 + 5.70e5T + 1.94e11T^{2} \)
43 \( 1 - 4.68e5T + 2.71e11T^{2} \)
47 \( 1 + 4.51e5T + 5.06e11T^{2} \)
53 \( 1 + 1.09e6T + 1.17e12T^{2} \)
61 \( 1 + 1.17e6T + 3.14e12T^{2} \)
67 \( 1 + 1.24e6T + 6.06e12T^{2} \)
71 \( 1 - 2.90e6T + 9.09e12T^{2} \)
73 \( 1 - 1.95e6T + 1.10e13T^{2} \)
79 \( 1 - 5.29e5T + 1.92e13T^{2} \)
83 \( 1 - 7.53e6T + 2.71e13T^{2} \)
89 \( 1 - 6.07e6T + 4.42e13T^{2} \)
97 \( 1 + 2.74e6T + 8.07e13T^{2} \)
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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

−12.05174131666716996843498251674, −10.36480261121774163018309215779, −9.822244679467962522229413344613, −8.019859612598104353442938176948, −6.50977441963052487062992045352, −6.15768402100904953098921177554, −4.83190181106411984301735102614, −3.70320424174285079886641173245, −2.60294096911288449698725870448, −1.72667109364519435154698547922, 1.72667109364519435154698547922, 2.60294096911288449698725870448, 3.70320424174285079886641173245, 4.83190181106411984301735102614, 6.15768402100904953098921177554, 6.50977441963052487062992045352, 8.019859612598104353442938176948, 9.822244679467962522229413344613, 10.36480261121774163018309215779, 12.05174131666716996843498251674

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