Properties

Label 2-177-1.1-c9-0-56
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  − 43.5·2-s + 81·3-s + 1.38e3·4-s + 1.59e3·5-s − 3.52e3·6-s − 1.09e4·7-s − 3.80e4·8-s + 6.56e3·9-s − 6.95e4·10-s + 5.29e4·11-s + 1.12e5·12-s − 1.46e5·13-s + 4.77e5·14-s + 1.29e5·15-s + 9.48e5·16-s − 2.76e5·17-s − 2.85e5·18-s + 9.99e5·19-s + 2.21e6·20-s − 8.88e5·21-s − 2.30e6·22-s + 1.88e6·23-s − 3.08e6·24-s + 5.97e5·25-s + 6.36e6·26-s + 5.31e5·27-s − 1.51e7·28-s + ⋯
L(s)  = 1  − 1.92·2-s + 0.577·3-s + 2.70·4-s + 1.14·5-s − 1.11·6-s − 1.72·7-s − 3.28·8-s + 0.333·9-s − 2.20·10-s + 1.09·11-s + 1.56·12-s − 1.41·13-s + 3.32·14-s + 0.659·15-s + 3.61·16-s − 0.802·17-s − 0.641·18-s + 1.76·19-s + 3.09·20-s − 0.996·21-s − 2.09·22-s + 1.40·23-s − 1.89·24-s + 0.306·25-s + 2.73·26-s + 0.192·27-s − 4.67·28-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 \( 1 + 43.5T + 512T^{2} \)
5 \( 1 - 1.59e3T + 1.95e6T^{2} \)
7 \( 1 + 1.09e4T + 4.03e7T^{2} \)
11 \( 1 - 5.29e4T + 2.35e9T^{2} \)
13 \( 1 + 1.46e5T + 1.06e10T^{2} \)
17 \( 1 + 2.76e5T + 1.18e11T^{2} \)
19 \( 1 - 9.99e5T + 3.22e11T^{2} \)
23 \( 1 - 1.88e6T + 1.80e12T^{2} \)
29 \( 1 + 6.91e6T + 1.45e13T^{2} \)
31 \( 1 + 1.40e6T + 2.64e13T^{2} \)
37 \( 1 - 4.90e6T + 1.29e14T^{2} \)
41 \( 1 - 2.05e7T + 3.27e14T^{2} \)
43 \( 1 + 3.94e6T + 5.02e14T^{2} \)
47 \( 1 - 6.35e6T + 1.11e15T^{2} \)
53 \( 1 + 7.48e7T + 3.29e15T^{2} \)
61 \( 1 + 7.46e7T + 1.16e16T^{2} \)
67 \( 1 - 2.24e7T + 2.72e16T^{2} \)
71 \( 1 + 2.63e8T + 4.58e16T^{2} \)
73 \( 1 - 1.35e8T + 5.88e16T^{2} \)
79 \( 1 - 1.42e8T + 1.19e17T^{2} \)
83 \( 1 + 1.25e7T + 1.86e17T^{2} \)
89 \( 1 + 8.08e8T + 3.50e17T^{2} \)
97 \( 1 + 1.04e8T + 7.60e17T^{2} \)
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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

−9.760846599090938836168448345924, −9.468847488232944322146925555272, −9.148466021626091326529187259007, −7.40188512720466432563867313212, −6.85097604105003971375788435351, −5.82646350133762608779865384263, −3.20765523685680322700813947676, −2.37589028400043394700527658283, −1.24601632531989112168078148230, 0, 1.24601632531989112168078148230, 2.37589028400043394700527658283, 3.20765523685680322700813947676, 5.82646350133762608779865384263, 6.85097604105003971375788435351, 7.40188512720466432563867313212, 9.148466021626091326529187259007, 9.468847488232944322146925555272, 9.760846599090938836168448345924

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