Properties

Label 2-177-1.1-c11-0-69
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  − 68.7·2-s + 243·3-s + 2.67e3·4-s + 7.32e3·5-s − 1.67e4·6-s − 6.39e4·7-s − 4.34e4·8-s + 5.90e4·9-s − 5.03e5·10-s − 3.07e5·11-s + 6.51e5·12-s − 1.14e6·13-s + 4.39e6·14-s + 1.77e6·15-s − 2.50e6·16-s + 1.41e6·17-s − 4.06e6·18-s + 4.00e6·19-s + 1.96e7·20-s − 1.55e7·21-s + 2.11e7·22-s + 4.77e7·23-s − 1.05e7·24-s + 4.82e6·25-s + 7.85e7·26-s + 1.43e7·27-s − 1.71e8·28-s + ⋯
L(s)  = 1  − 1.51·2-s + 0.577·3-s + 1.30·4-s + 1.04·5-s − 0.877·6-s − 1.43·7-s − 0.468·8-s + 0.333·9-s − 1.59·10-s − 0.576·11-s + 0.755·12-s − 0.853·13-s + 2.18·14-s + 0.605·15-s − 0.596·16-s + 0.242·17-s − 0.506·18-s + 0.370·19-s + 1.37·20-s − 0.830·21-s + 0.875·22-s + 1.54·23-s − 0.270·24-s + 0.0987·25-s + 1.29·26-s + 0.192·27-s − 1.88·28-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 \( 1 + 68.7T + 2.04e3T^{2} \)
5 \( 1 - 7.32e3T + 4.88e7T^{2} \)
7 \( 1 + 6.39e4T + 1.97e9T^{2} \)
11 \( 1 + 3.07e5T + 2.85e11T^{2} \)
13 \( 1 + 1.14e6T + 1.79e12T^{2} \)
17 \( 1 - 1.41e6T + 3.42e13T^{2} \)
19 \( 1 - 4.00e6T + 1.16e14T^{2} \)
23 \( 1 - 4.77e7T + 9.52e14T^{2} \)
29 \( 1 - 1.63e8T + 1.22e16T^{2} \)
31 \( 1 - 1.17e8T + 2.54e16T^{2} \)
37 \( 1 + 5.45e8T + 1.77e17T^{2} \)
41 \( 1 + 5.77e8T + 5.50e17T^{2} \)
43 \( 1 + 1.16e9T + 9.29e17T^{2} \)
47 \( 1 + 3.29e8T + 2.47e18T^{2} \)
53 \( 1 - 5.23e9T + 9.26e18T^{2} \)
61 \( 1 - 6.69e9T + 4.35e19T^{2} \)
67 \( 1 + 1.09e10T + 1.22e20T^{2} \)
71 \( 1 + 1.13e10T + 2.31e20T^{2} \)
73 \( 1 + 1.17e10T + 3.13e20T^{2} \)
79 \( 1 - 4.67e10T + 7.47e20T^{2} \)
83 \( 1 + 4.41e9T + 1.28e21T^{2} \)
89 \( 1 - 2.79e9T + 2.77e21T^{2} \)
97 \( 1 + 1.07e11T + 7.15e21T^{2} \)
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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

−10.09595902445133490461409900018, −9.267734451986253444772339114876, −8.481686023278337432804057197924, −7.21500536310999736373474383591, −6.54295359203447116310606518326, −5.08290286006062986833085989133, −3.10643477187431478911419753517, −2.34916784492382513481763346825, −1.12306926627456847815940503238, 0, 1.12306926627456847815940503238, 2.34916784492382513481763346825, 3.10643477187431478911419753517, 5.08290286006062986833085989133, 6.54295359203447116310606518326, 7.21500536310999736373474383591, 8.481686023278337432804057197924, 9.267734451986253444772339114876, 10.09595902445133490461409900018

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