Properties

Label 2-177-1.1-c7-0-5
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  − 8.14·2-s − 27·3-s − 61.6·4-s − 54.3·5-s + 219.·6-s − 413.·7-s + 1.54e3·8-s + 729·9-s + 442.·10-s + 5.30e3·11-s + 1.66e3·12-s − 1.34e4·13-s + 3.36e3·14-s + 1.46e3·15-s − 4.68e3·16-s + 1.37e4·17-s − 5.93e3·18-s − 5.78e4·19-s + 3.34e3·20-s + 1.11e4·21-s − 4.31e4·22-s − 1.18e4·23-s − 4.17e4·24-s − 7.51e4·25-s + 1.09e5·26-s − 1.96e4·27-s + 2.54e4·28-s + ⋯
L(s)  = 1  − 0.719·2-s − 0.577·3-s − 0.481·4-s − 0.194·5-s + 0.415·6-s − 0.455·7-s + 1.06·8-s + 0.333·9-s + 0.139·10-s + 1.20·11-s + 0.278·12-s − 1.70·13-s + 0.327·14-s + 0.112·15-s − 0.286·16-s + 0.677·17-s − 0.239·18-s − 1.93·19-s + 0.0936·20-s + 0.263·21-s − 0.864·22-s − 0.202·23-s − 0.615·24-s − 0.962·25-s + 1.22·26-s − 0.192·27-s + 0.219·28-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

\(L(4)\) \(\approx\) \(0.3865655091\)
\(L(\frac12)\) \(\approx\) \(0.3865655091\)
\(L(\frac{9}{2})\) 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 \( 1 + 8.14T + 128T^{2} \)
5 \( 1 + 54.3T + 7.81e4T^{2} \)
7 \( 1 + 413.T + 8.23e5T^{2} \)
11 \( 1 - 5.30e3T + 1.94e7T^{2} \)
13 \( 1 + 1.34e4T + 6.27e7T^{2} \)
17 \( 1 - 1.37e4T + 4.10e8T^{2} \)
19 \( 1 + 5.78e4T + 8.93e8T^{2} \)
23 \( 1 + 1.18e4T + 3.40e9T^{2} \)
29 \( 1 + 1.37e5T + 1.72e10T^{2} \)
31 \( 1 - 1.10e5T + 2.75e10T^{2} \)
37 \( 1 - 1.54e5T + 9.49e10T^{2} \)
41 \( 1 - 5.80e4T + 1.94e11T^{2} \)
43 \( 1 + 6.07e5T + 2.71e11T^{2} \)
47 \( 1 + 7.65e5T + 5.06e11T^{2} \)
53 \( 1 - 7.79e5T + 1.17e12T^{2} \)
61 \( 1 - 7.45e5T + 3.14e12T^{2} \)
67 \( 1 + 3.43e5T + 6.06e12T^{2} \)
71 \( 1 - 4.41e6T + 9.09e12T^{2} \)
73 \( 1 - 1.38e6T + 1.10e13T^{2} \)
79 \( 1 + 6.92e6T + 1.92e13T^{2} \)
83 \( 1 - 4.41e6T + 2.71e13T^{2} \)
89 \( 1 + 2.65e6T + 4.42e13T^{2} \)
97 \( 1 - 5.31e5T + 8.07e13T^{2} \)
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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

−11.33189101565619708685649411226, −10.03249900986698028099219504334, −9.630361344274275621266727021271, −8.423412579888918354627656464264, −7.34750232677267397230710689736, −6.28127056333566503958068225719, −4.85109760222685174633947410138, −3.87656816244684727518303632584, −1.87925526490131357501477134275, −0.38786162398710739281472581645, 0.38786162398710739281472581645, 1.87925526490131357501477134275, 3.87656816244684727518303632584, 4.85109760222685174633947410138, 6.28127056333566503958068225719, 7.34750232677267397230710689736, 8.423412579888918354627656464264, 9.630361344274275621266727021271, 10.03249900986698028099219504334, 11.33189101565619708685649411226

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