Properties

Label 2-177-1.1-c7-0-56
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 + 347.·4-s + 424.·5-s − 588.·6-s + 1.19e3·7-s + 4.79e3·8-s + 729·9-s + 9.26e3·10-s − 3.09e3·11-s − 9.39e3·12-s + 1.04e3·13-s + 2.60e4·14-s − 1.14e4·15-s + 6.00e4·16-s − 1.66e4·17-s + 1.59e4·18-s + 2.55e4·19-s + 1.47e5·20-s − 3.22e4·21-s − 6.74e4·22-s − 9.87e4·23-s − 1.29e5·24-s + 1.02e5·25-s + 2.27e4·26-s − 1.96e4·27-s + 4.14e5·28-s + ⋯
L(s)  = 1  + 1.92·2-s − 0.577·3-s + 2.71·4-s + 1.52·5-s − 1.11·6-s + 1.31·7-s + 3.31·8-s + 0.333·9-s + 2.93·10-s − 0.700·11-s − 1.56·12-s + 0.131·13-s + 2.53·14-s − 0.877·15-s + 3.66·16-s − 0.820·17-s + 0.642·18-s + 0.855·19-s + 4.13·20-s − 0.758·21-s − 1.35·22-s − 1.69·23-s − 1.91·24-s + 1.31·25-s + 0.254·26-s − 0.192·27-s + 3.57·28-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

\(L(4)\) \(\approx\) \(9.199812643\)
\(L(\frac12)\) \(\approx\) \(9.199812643\)
\(L(\frac{9}{2})\) 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 \( 1 - 424.T + 7.81e4T^{2} \)
7 \( 1 - 1.19e3T + 8.23e5T^{2} \)
11 \( 1 + 3.09e3T + 1.94e7T^{2} \)
13 \( 1 - 1.04e3T + 6.27e7T^{2} \)
17 \( 1 + 1.66e4T + 4.10e8T^{2} \)
19 \( 1 - 2.55e4T + 8.93e8T^{2} \)
23 \( 1 + 9.87e4T + 3.40e9T^{2} \)
29 \( 1 + 2.41e5T + 1.72e10T^{2} \)
31 \( 1 + 1.17e5T + 2.75e10T^{2} \)
37 \( 1 - 7.34e4T + 9.49e10T^{2} \)
41 \( 1 + 3.33e5T + 1.94e11T^{2} \)
43 \( 1 - 6.08e5T + 2.71e11T^{2} \)
47 \( 1 + 9.23e5T + 5.06e11T^{2} \)
53 \( 1 - 6.17e5T + 1.17e12T^{2} \)
61 \( 1 - 3.02e6T + 3.14e12T^{2} \)
67 \( 1 - 4.24e6T + 6.06e12T^{2} \)
71 \( 1 - 8.09e5T + 9.09e12T^{2} \)
73 \( 1 - 1.31e6T + 1.10e13T^{2} \)
79 \( 1 - 5.69e6T + 1.92e13T^{2} \)
83 \( 1 + 3.55e6T + 2.71e13T^{2} \)
89 \( 1 + 1.13e7T + 4.42e13T^{2} \)
97 \( 1 + 3.73e6T + 8.07e13T^{2} \)
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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

−11.47121142296658693599483848291, −10.93739922461963767146489576749, −9.846278031941274919398867895475, −7.85184122277972456628283315301, −6.65995132698684885293295818356, −5.49009809986635294350764448399, −5.33392391850219914261465509094, −4.02781204427356416404371460246, −2.27603174719015657520765618953, −1.66709054492680548484409165125, 1.66709054492680548484409165125, 2.27603174719015657520765618953, 4.02781204427356416404371460246, 5.33392391850219914261465509094, 5.49009809986635294350764448399, 6.65995132698684885293295818356, 7.85184122277972456628283315301, 9.846278031941274919398867895475, 10.93739922461963767146489576749, 11.47121142296658693599483848291

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