Properties

Label 2-177-1.1-c7-0-43
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  + 7.02·2-s − 27·3-s − 78.6·4-s − 266.·5-s − 189.·6-s + 665.·7-s − 1.45e3·8-s + 729·9-s − 1.87e3·10-s + 2.99e3·11-s + 2.12e3·12-s + 1.11e4·13-s + 4.67e3·14-s + 7.20e3·15-s − 120.·16-s + 1.73e4·17-s + 5.11e3·18-s − 7.56e3·19-s + 2.09e4·20-s − 1.79e4·21-s + 2.10e4·22-s − 8.48e4·23-s + 3.91e4·24-s − 6.95e3·25-s + 7.80e4·26-s − 1.96e4·27-s − 5.23e4·28-s + ⋯
L(s)  = 1  + 0.620·2-s − 0.577·3-s − 0.614·4-s − 0.954·5-s − 0.358·6-s + 0.733·7-s − 1.00·8-s + 0.333·9-s − 0.592·10-s + 0.678·11-s + 0.354·12-s + 1.40·13-s + 0.455·14-s + 0.551·15-s − 0.00733·16-s + 0.855·17-s + 0.206·18-s − 0.253·19-s + 0.586·20-s − 0.423·21-s + 0.421·22-s − 1.45·23-s + 0.578·24-s − 0.0890·25-s + 0.870·26-s − 0.192·27-s − 0.450·28-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 \( 1 - 7.02T + 128T^{2} \)
5 \( 1 + 266.T + 7.81e4T^{2} \)
7 \( 1 - 665.T + 8.23e5T^{2} \)
11 \( 1 - 2.99e3T + 1.94e7T^{2} \)
13 \( 1 - 1.11e4T + 6.27e7T^{2} \)
17 \( 1 - 1.73e4T + 4.10e8T^{2} \)
19 \( 1 + 7.56e3T + 8.93e8T^{2} \)
23 \( 1 + 8.48e4T + 3.40e9T^{2} \)
29 \( 1 - 5.97e4T + 1.72e10T^{2} \)
31 \( 1 + 5.62e4T + 2.75e10T^{2} \)
37 \( 1 + 3.75e5T + 9.49e10T^{2} \)
41 \( 1 - 5.60e5T + 1.94e11T^{2} \)
43 \( 1 + 5.99e5T + 2.71e11T^{2} \)
47 \( 1 + 5.08e5T + 5.06e11T^{2} \)
53 \( 1 - 9.51e4T + 1.17e12T^{2} \)
61 \( 1 - 1.08e6T + 3.14e12T^{2} \)
67 \( 1 + 7.31e5T + 6.06e12T^{2} \)
71 \( 1 - 4.30e6T + 9.09e12T^{2} \)
73 \( 1 + 8.04e5T + 1.10e13T^{2} \)
79 \( 1 + 3.53e6T + 1.92e13T^{2} \)
83 \( 1 + 8.87e6T + 2.71e13T^{2} \)
89 \( 1 + 5.83e6T + 4.42e13T^{2} \)
97 \( 1 - 1.47e7T + 8.07e13T^{2} \)
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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

−11.29457124482352558825937177982, −10.01693090855648701576993789383, −8.655698597064306907605014713134, −7.922315510076863896815162714471, −6.39592128582760935861291458078, −5.39749936419377176210052277549, −4.20782567262960017636619195105, −3.60362322088298195603989508758, −1.32950575629041538734998936575, 0, 1.32950575629041538734998936575, 3.60362322088298195603989508758, 4.20782567262960017636619195105, 5.39749936419377176210052277549, 6.39592128582760935861291458078, 7.922315510076863896815162714471, 8.655698597064306907605014713134, 10.01693090855648701576993789383, 11.29457124482352558825937177982

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