Properties

Label 2-177-1.1-c11-0-65
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  − 16.5·2-s − 243·3-s − 1.77e3·4-s + 2.84e3·5-s + 4.02e3·6-s + 4.14e4·7-s + 6.32e4·8-s + 5.90e4·9-s − 4.70e4·10-s + 4.94e4·11-s + 4.31e5·12-s − 2.15e6·13-s − 6.85e5·14-s − 6.91e5·15-s + 2.58e6·16-s + 2.16e6·17-s − 9.77e5·18-s + 7.65e6·19-s − 5.04e6·20-s − 1.00e7·21-s − 8.17e5·22-s − 2.58e7·23-s − 1.53e7·24-s − 4.07e7·25-s + 3.57e7·26-s − 1.43e7·27-s − 7.35e7·28-s + ⋯
L(s)  = 1  − 0.365·2-s − 0.577·3-s − 0.866·4-s + 0.407·5-s + 0.211·6-s + 0.932·7-s + 0.682·8-s + 0.333·9-s − 0.148·10-s + 0.0924·11-s + 0.500·12-s − 1.61·13-s − 0.340·14-s − 0.235·15-s + 0.616·16-s + 0.370·17-s − 0.121·18-s + 0.709·19-s − 0.352·20-s − 0.538·21-s − 0.0338·22-s − 0.835·23-s − 0.393·24-s − 0.834·25-s + 0.589·26-s − 0.192·27-s − 0.807·28-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 \( 1 + 16.5T + 2.04e3T^{2} \)
5 \( 1 - 2.84e3T + 4.88e7T^{2} \)
7 \( 1 - 4.14e4T + 1.97e9T^{2} \)
11 \( 1 - 4.94e4T + 2.85e11T^{2} \)
13 \( 1 + 2.15e6T + 1.79e12T^{2} \)
17 \( 1 - 2.16e6T + 3.42e13T^{2} \)
19 \( 1 - 7.65e6T + 1.16e14T^{2} \)
23 \( 1 + 2.58e7T + 9.52e14T^{2} \)
29 \( 1 + 5.29e7T + 1.22e16T^{2} \)
31 \( 1 + 7.09e6T + 2.54e16T^{2} \)
37 \( 1 - 1.02e7T + 1.77e17T^{2} \)
41 \( 1 - 1.31e9T + 5.50e17T^{2} \)
43 \( 1 - 1.41e9T + 9.29e17T^{2} \)
47 \( 1 + 9.75e8T + 2.47e18T^{2} \)
53 \( 1 + 9.42e8T + 9.26e18T^{2} \)
61 \( 1 - 7.22e9T + 4.35e19T^{2} \)
67 \( 1 - 1.55e10T + 1.22e20T^{2} \)
71 \( 1 - 1.91e10T + 2.31e20T^{2} \)
73 \( 1 + 1.37e10T + 3.13e20T^{2} \)
79 \( 1 + 3.29e9T + 7.47e20T^{2} \)
83 \( 1 + 6.18e10T + 1.28e21T^{2} \)
89 \( 1 - 7.94e10T + 2.77e21T^{2} \)
97 \( 1 + 1.08e11T + 7.15e21T^{2} \)
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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

−9.909891297152952218682771848859, −9.466820418074655559812052947492, −8.047981033437206782190500298491, −7.37830371344171785003917601482, −5.73137943305444842325593484839, −4.99794124934209157588916210208, −4.05021435930305688210514468545, −2.22049820437803305610406186136, −1.07947872064870151209112154280, 0, 1.07947872064870151209112154280, 2.22049820437803305610406186136, 4.05021435930305688210514468545, 4.99794124934209157588916210208, 5.73137943305444842325593484839, 7.37830371344171785003917601482, 8.047981033437206782190500298491, 9.466820418074655559812052947492, 9.909891297152952218682771848859

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