Properties

Label 2-177-1.1-c7-0-41
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.00·2-s − 27·3-s − 78.8·4-s + 449.·5-s + 189.·6-s − 271.·7-s + 1.44e3·8-s + 729·9-s − 3.14e3·10-s − 3.16e3·11-s + 2.12e3·12-s − 9.44e3·13-s + 1.90e3·14-s − 1.21e4·15-s − 63.3·16-s − 6.79e3·17-s − 5.10e3·18-s + 3.40e4·19-s − 3.54e4·20-s + 7.33e3·21-s + 2.21e4·22-s − 3.12e4·23-s − 3.91e4·24-s + 1.23e5·25-s + 6.62e4·26-s − 1.96e4·27-s + 2.14e4·28-s + ⋯
L(s)  = 1  − 0.619·2-s − 0.577·3-s − 0.616·4-s + 1.60·5-s + 0.357·6-s − 0.299·7-s + 1.00·8-s + 0.333·9-s − 0.995·10-s − 0.716·11-s + 0.355·12-s − 1.19·13-s + 0.185·14-s − 0.927·15-s − 0.00386·16-s − 0.335·17-s − 0.206·18-s + 1.14·19-s − 0.990·20-s + 0.172·21-s + 0.444·22-s − 0.535·23-s − 0.578·24-s + 1.58·25-s + 0.738·26-s − 0.192·27-s + 0.184·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 + 7.00T + 128T^{2} \)
5 \( 1 - 449.T + 7.81e4T^{2} \)
7 \( 1 + 271.T + 8.23e5T^{2} \)
11 \( 1 + 3.16e3T + 1.94e7T^{2} \)
13 \( 1 + 9.44e3T + 6.27e7T^{2} \)
17 \( 1 + 6.79e3T + 4.10e8T^{2} \)
19 \( 1 - 3.40e4T + 8.93e8T^{2} \)
23 \( 1 + 3.12e4T + 3.40e9T^{2} \)
29 \( 1 - 8.27e4T + 1.72e10T^{2} \)
31 \( 1 - 2.83e5T + 2.75e10T^{2} \)
37 \( 1 - 2.68e4T + 9.49e10T^{2} \)
41 \( 1 + 1.95e4T + 1.94e11T^{2} \)
43 \( 1 + 4.17e5T + 2.71e11T^{2} \)
47 \( 1 + 2.76e4T + 5.06e11T^{2} \)
53 \( 1 - 2.90e4T + 1.17e12T^{2} \)
61 \( 1 - 1.53e6T + 3.14e12T^{2} \)
67 \( 1 + 2.52e6T + 6.06e12T^{2} \)
71 \( 1 + 3.35e6T + 9.09e12T^{2} \)
73 \( 1 + 1.90e6T + 1.10e13T^{2} \)
79 \( 1 - 2.26e6T + 1.92e13T^{2} \)
83 \( 1 + 5.96e6T + 2.71e13T^{2} \)
89 \( 1 - 5.11e6T + 4.42e13T^{2} \)
97 \( 1 + 1.45e5T + 8.07e13T^{2} \)
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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

−10.25173296399467714103233141388, −10.05110246321336695879843320858, −9.197810110828452276931190318827, −7.893561643546774666199574023448, −6.64097647166103652380424443356, −5.46760923706317448935085144831, −4.72833963720450325788419455087, −2.63562009327498367068622249221, −1.30053335634186982595138684331, 0, 1.30053335634186982595138684331, 2.63562009327498367068622249221, 4.72833963720450325788419455087, 5.46760923706317448935085144831, 6.64097647166103652380424443356, 7.893561643546774666199574023448, 9.197810110828452276931190318827, 10.05110246321336695879843320858, 10.25173296399467714103233141388

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