Properties

Label 2-177-1.1-c9-0-6
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13.7·2-s − 81·3-s − 324.·4-s − 1.28e3·5-s + 1.11e3·6-s + 3.21e3·7-s + 1.14e4·8-s + 6.56e3·9-s + 1.76e4·10-s − 5.83e4·11-s + 2.62e4·12-s + 1.42e5·13-s − 4.41e4·14-s + 1.04e5·15-s + 8.86e3·16-s − 6.34e5·17-s − 8.99e4·18-s + 1.07e6·19-s + 4.17e5·20-s − 2.60e5·21-s + 7.99e5·22-s − 3.25e5·23-s − 9.28e5·24-s − 2.94e5·25-s − 1.95e6·26-s − 5.31e5·27-s − 1.04e6·28-s + ⋯
L(s)  = 1  − 0.605·2-s − 0.577·3-s − 0.633·4-s − 0.921·5-s + 0.349·6-s + 0.506·7-s + 0.989·8-s + 0.333·9-s + 0.558·10-s − 1.20·11-s + 0.365·12-s + 1.38·13-s − 0.306·14-s + 0.532·15-s + 0.0338·16-s − 1.84·17-s − 0.201·18-s + 1.88·19-s + 0.583·20-s − 0.292·21-s + 0.727·22-s − 0.242·23-s − 0.571·24-s − 0.150·25-s − 0.839·26-s − 0.192·27-s − 0.320·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.3899076537\)
\(L(\frac12)\) \(\approx\) \(0.3899076537\)
\(L(\frac{11}{2})\) 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 \( 1 + 13.7T + 512T^{2} \)
5 \( 1 + 1.28e3T + 1.95e6T^{2} \)
7 \( 1 - 3.21e3T + 4.03e7T^{2} \)
11 \( 1 + 5.83e4T + 2.35e9T^{2} \)
13 \( 1 - 1.42e5T + 1.06e10T^{2} \)
17 \( 1 + 6.34e5T + 1.18e11T^{2} \)
19 \( 1 - 1.07e6T + 3.22e11T^{2} \)
23 \( 1 + 3.25e5T + 1.80e12T^{2} \)
29 \( 1 + 4.66e6T + 1.45e13T^{2} \)
31 \( 1 + 5.51e6T + 2.64e13T^{2} \)
37 \( 1 - 1.62e7T + 1.29e14T^{2} \)
41 \( 1 + 2.02e7T + 3.27e14T^{2} \)
43 \( 1 + 3.84e7T + 5.02e14T^{2} \)
47 \( 1 + 3.59e6T + 1.11e15T^{2} \)
53 \( 1 - 5.97e7T + 3.29e15T^{2} \)
61 \( 1 - 5.83e7T + 1.16e16T^{2} \)
67 \( 1 - 1.77e7T + 2.72e16T^{2} \)
71 \( 1 + 1.90e8T + 4.58e16T^{2} \)
73 \( 1 + 4.70e8T + 5.88e16T^{2} \)
79 \( 1 + 1.31e8T + 1.19e17T^{2} \)
83 \( 1 + 5.84e7T + 1.86e17T^{2} \)
89 \( 1 - 5.00e8T + 3.50e17T^{2} \)
97 \( 1 - 5.11e8T + 7.60e17T^{2} \)
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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

−11.16123893431053354140192342177, −10.05840432238839689240206349657, −8.870844765286802037011236111921, −8.029922582840290404141364324692, −7.23707499208858417159593949925, −5.60300641742134225401241013131, −4.63237004438984067434644815502, −3.59114610158747880728115843431, −1.62356764506891765591320638164, −0.36053967947958405281710988710, 0.36053967947958405281710988710, 1.62356764506891765591320638164, 3.59114610158747880728115843431, 4.63237004438984067434644815502, 5.60300641742134225401241013131, 7.23707499208858417159593949925, 8.029922582840290404141364324692, 8.870844765286802037011236111921, 10.05840432238839689240206349657, 11.16123893431053354140192342177

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