Properties

Label 2-177-1.1-c11-0-33
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16.4·2-s − 243·3-s − 1.77e3·4-s − 1.11e4·5-s − 3.99e3·6-s − 6.47e4·7-s − 6.29e4·8-s + 5.90e4·9-s − 1.83e5·10-s − 1.52e5·11-s + 4.31e5·12-s + 2.10e6·13-s − 1.06e6·14-s + 2.71e6·15-s + 2.60e6·16-s − 1.11e7·17-s + 9.71e5·18-s − 1.27e7·19-s + 1.98e7·20-s + 1.57e7·21-s − 2.51e6·22-s + 9.67e6·23-s + 1.52e7·24-s + 7.59e7·25-s + 3.47e7·26-s − 1.43e7·27-s + 1.15e8·28-s + ⋯
L(s)  = 1  + 0.363·2-s − 0.577·3-s − 0.867·4-s − 1.59·5-s − 0.209·6-s − 1.45·7-s − 0.679·8-s + 0.333·9-s − 0.581·10-s − 0.286·11-s + 0.500·12-s + 1.57·13-s − 0.529·14-s + 0.922·15-s + 0.620·16-s − 1.90·17-s + 0.121·18-s − 1.18·19-s + 1.38·20-s + 0.840·21-s − 0.104·22-s + 0.313·23-s + 0.392·24-s + 1.55·25-s + 0.573·26-s − 0.192·27-s + 1.26·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 243T \)
59 \( 1 - 7.14e8T \)
good2 \( 1 - 16.4T + 2.04e3T^{2} \)
5 \( 1 + 1.11e4T + 4.88e7T^{2} \)
7 \( 1 + 6.47e4T + 1.97e9T^{2} \)
11 \( 1 + 1.52e5T + 2.85e11T^{2} \)
13 \( 1 - 2.10e6T + 1.79e12T^{2} \)
17 \( 1 + 1.11e7T + 3.42e13T^{2} \)
19 \( 1 + 1.27e7T + 1.16e14T^{2} \)
23 \( 1 - 9.67e6T + 9.52e14T^{2} \)
29 \( 1 - 1.78e8T + 1.22e16T^{2} \)
31 \( 1 - 8.93e7T + 2.54e16T^{2} \)
37 \( 1 + 2.78e8T + 1.77e17T^{2} \)
41 \( 1 - 9.14e8T + 5.50e17T^{2} \)
43 \( 1 + 8.43e7T + 9.29e17T^{2} \)
47 \( 1 + 1.29e9T + 2.47e18T^{2} \)
53 \( 1 - 3.97e9T + 9.26e18T^{2} \)
61 \( 1 + 6.12e9T + 4.35e19T^{2} \)
67 \( 1 - 5.63e9T + 1.22e20T^{2} \)
71 \( 1 + 1.19e10T + 2.31e20T^{2} \)
73 \( 1 + 2.52e10T + 3.13e20T^{2} \)
79 \( 1 - 4.83e10T + 7.47e20T^{2} \)
83 \( 1 + 6.79e10T + 1.28e21T^{2} \)
89 \( 1 - 4.73e10T + 2.77e21T^{2} \)
97 \( 1 + 1.10e11T + 7.15e21T^{2} \)
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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

−10.37728815888642686400883858124, −8.922989648689079075116081755207, −8.372381206599976001427373150446, −6.80046812611224523408503732140, −6.12799709578067003027560915015, −4.52519970997428366096835051871, −3.99568306176289067551321785481, −3.00153850347900188835458722677, −0.68399124128263829123891802870, 0, 0.68399124128263829123891802870, 3.00153850347900188835458722677, 3.99568306176289067551321785481, 4.52519970997428366096835051871, 6.12799709578067003027560915015, 6.80046812611224523408503732140, 8.372381206599976001427373150446, 8.922989648689079075116081755207, 10.37728815888642686400883858124

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