Properties

Label 2-177-1.1-c7-0-24
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  − 19.7·2-s − 27·3-s + 261.·4-s − 90.5·5-s + 532.·6-s − 807.·7-s − 2.63e3·8-s + 729·9-s + 1.78e3·10-s + 4.97e3·11-s − 7.06e3·12-s − 2.92e3·13-s + 1.59e4·14-s + 2.44e3·15-s + 1.85e4·16-s − 2.93e4·17-s − 1.43e4·18-s + 8.90e3·19-s − 2.36e4·20-s + 2.18e4·21-s − 9.81e4·22-s − 2.13e4·23-s + 7.11e4·24-s − 6.99e4·25-s + 5.76e4·26-s − 1.96e4·27-s − 2.11e5·28-s + ⋯
L(s)  = 1  − 1.74·2-s − 0.577·3-s + 2.04·4-s − 0.324·5-s + 1.00·6-s − 0.890·7-s − 1.81·8-s + 0.333·9-s + 0.565·10-s + 1.12·11-s − 1.17·12-s − 0.368·13-s + 1.55·14-s + 0.187·15-s + 1.13·16-s − 1.44·17-s − 0.581·18-s + 0.297·19-s − 0.662·20-s + 0.513·21-s − 1.96·22-s − 0.365·23-s + 1.05·24-s − 0.894·25-s + 0.643·26-s − 0.192·27-s − 1.81·28-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 \( 1 + 19.7T + 128T^{2} \)
5 \( 1 + 90.5T + 7.81e4T^{2} \)
7 \( 1 + 807.T + 8.23e5T^{2} \)
11 \( 1 - 4.97e3T + 1.94e7T^{2} \)
13 \( 1 + 2.92e3T + 6.27e7T^{2} \)
17 \( 1 + 2.93e4T + 4.10e8T^{2} \)
19 \( 1 - 8.90e3T + 8.93e8T^{2} \)
23 \( 1 + 2.13e4T + 3.40e9T^{2} \)
29 \( 1 - 1.44e5T + 1.72e10T^{2} \)
31 \( 1 - 1.24e5T + 2.75e10T^{2} \)
37 \( 1 - 2.49e5T + 9.49e10T^{2} \)
41 \( 1 - 7.69e5T + 1.94e11T^{2} \)
43 \( 1 + 2.14e5T + 2.71e11T^{2} \)
47 \( 1 - 8.26e5T + 5.06e11T^{2} \)
53 \( 1 - 7.91e5T + 1.17e12T^{2} \)
61 \( 1 + 2.23e6T + 3.14e12T^{2} \)
67 \( 1 + 6.01e4T + 6.06e12T^{2} \)
71 \( 1 - 3.73e6T + 9.09e12T^{2} \)
73 \( 1 - 4.76e5T + 1.10e13T^{2} \)
79 \( 1 - 1.82e6T + 1.92e13T^{2} \)
83 \( 1 - 4.01e6T + 2.71e13T^{2} \)
89 \( 1 - 5.38e6T + 4.42e13T^{2} \)
97 \( 1 + 1.52e7T + 8.07e13T^{2} \)
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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

−10.71042576315969463079689493879, −9.693924085864529131754711145945, −9.113769986859210026588758178488, −7.924951495822583243516048459259, −6.81845557333484972922038763169, −6.22405931252589715892827833297, −4.19753475606811314431755752728, −2.45414143509716166340650843244, −0.990043362036185300131713793310, 0, 0.990043362036185300131713793310, 2.45414143509716166340650843244, 4.19753475606811314431755752728, 6.22405931252589715892827833297, 6.81845557333484972922038763169, 7.924951495822583243516048459259, 9.113769986859210026588758178488, 9.693924085864529131754711145945, 10.71042576315969463079689493879

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