Properties

Label 2-177-1.1-c9-0-49
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 43.9·2-s + 81·3-s + 1.41e3·4-s − 993.·5-s − 3.55e3·6-s + 3.42e3·7-s − 3.97e4·8-s + 6.56e3·9-s + 4.36e4·10-s − 7.50e4·11-s + 1.14e5·12-s + 6.68e4·13-s − 1.50e5·14-s − 8.04e4·15-s + 1.01e6·16-s + 1.99e5·17-s − 2.88e5·18-s + 7.98e4·19-s − 1.40e6·20-s + 2.77e5·21-s + 3.29e6·22-s + 2.06e6·23-s − 3.21e6·24-s − 9.65e5·25-s − 2.93e6·26-s + 5.31e5·27-s + 4.85e6·28-s + ⋯
L(s)  = 1  − 1.94·2-s + 0.577·3-s + 2.76·4-s − 0.710·5-s − 1.12·6-s + 0.539·7-s − 3.42·8-s + 0.333·9-s + 1.37·10-s − 1.54·11-s + 1.59·12-s + 0.649·13-s − 1.04·14-s − 0.410·15-s + 3.88·16-s + 0.579·17-s − 0.646·18-s + 0.140·19-s − 1.96·20-s + 0.311·21-s + 3.00·22-s + 1.53·23-s − 1.97·24-s − 0.494·25-s − 1.26·26-s + 0.192·27-s + 1.49·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 81T \)
59 \( 1 - 1.21e7T \)
good2 \( 1 + 43.9T + 512T^{2} \)
5 \( 1 + 993.T + 1.95e6T^{2} \)
7 \( 1 - 3.42e3T + 4.03e7T^{2} \)
11 \( 1 + 7.50e4T + 2.35e9T^{2} \)
13 \( 1 - 6.68e4T + 1.06e10T^{2} \)
17 \( 1 - 1.99e5T + 1.18e11T^{2} \)
19 \( 1 - 7.98e4T + 3.22e11T^{2} \)
23 \( 1 - 2.06e6T + 1.80e12T^{2} \)
29 \( 1 + 5.68e6T + 1.45e13T^{2} \)
31 \( 1 - 1.44e6T + 2.64e13T^{2} \)
37 \( 1 + 6.74e6T + 1.29e14T^{2} \)
41 \( 1 + 3.10e7T + 3.27e14T^{2} \)
43 \( 1 - 3.99e7T + 5.02e14T^{2} \)
47 \( 1 - 1.76e7T + 1.11e15T^{2} \)
53 \( 1 - 9.14e7T + 3.29e15T^{2} \)
61 \( 1 + 3.45e7T + 1.16e16T^{2} \)
67 \( 1 - 4.20e7T + 2.72e16T^{2} \)
71 \( 1 - 2.88e8T + 4.58e16T^{2} \)
73 \( 1 + 4.61e8T + 5.88e16T^{2} \)
79 \( 1 + 3.32e6T + 1.19e17T^{2} \)
83 \( 1 - 4.97e8T + 1.86e17T^{2} \)
89 \( 1 - 5.53e8T + 3.50e17T^{2} \)
97 \( 1 + 4.40e8T + 7.60e17T^{2} \)
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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

−10.44755886772608185990956372730, −9.315003990342768542457619820726, −8.409581232044859424212407731590, −7.78050817225953369528589513410, −7.12180089745040088697708533172, −5.48532259908728932985461746995, −3.40960867608328551621351779029, −2.31978706073555138836777960943, −1.14246781129545837827254691319, 0, 1.14246781129545837827254691319, 2.31978706073555138836777960943, 3.40960867608328551621351779029, 5.48532259908728932985461746995, 7.12180089745040088697708533172, 7.78050817225953369528589513410, 8.409581232044859424212407731590, 9.315003990342768542457619820726, 10.44755886772608185990956372730

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