Properties

Label 2-177-1.1-c11-0-98
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  + 87.0·2-s − 243·3-s + 5.53e3·4-s − 4.10e3·5-s − 2.11e4·6-s − 4.94e4·7-s + 3.03e5·8-s + 5.90e4·9-s − 3.57e5·10-s + 6.02e4·11-s − 1.34e6·12-s + 4.34e5·13-s − 4.30e6·14-s + 9.97e5·15-s + 1.51e7·16-s − 9.38e6·17-s + 5.14e6·18-s + 1.79e7·19-s − 2.27e7·20-s + 1.20e7·21-s + 5.24e6·22-s + 4.13e7·23-s − 7.37e7·24-s − 3.19e7·25-s + 3.77e7·26-s − 1.43e7·27-s − 2.73e8·28-s + ⋯
L(s)  = 1  + 1.92·2-s − 0.577·3-s + 2.70·4-s − 0.587·5-s − 1.11·6-s − 1.11·7-s + 3.27·8-s + 0.333·9-s − 1.13·10-s + 0.112·11-s − 1.56·12-s + 0.324·13-s − 2.14·14-s + 0.339·15-s + 3.60·16-s − 1.60·17-s + 0.641·18-s + 1.66·19-s − 1.58·20-s + 0.642·21-s + 0.217·22-s + 1.33·23-s − 1.89·24-s − 0.655·25-s + 0.623·26-s − 0.192·27-s − 3.00·28-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 \( 1 - 87.0T + 2.04e3T^{2} \)
5 \( 1 + 4.10e3T + 4.88e7T^{2} \)
7 \( 1 + 4.94e4T + 1.97e9T^{2} \)
11 \( 1 - 6.02e4T + 2.85e11T^{2} \)
13 \( 1 - 4.34e5T + 1.79e12T^{2} \)
17 \( 1 + 9.38e6T + 3.42e13T^{2} \)
19 \( 1 - 1.79e7T + 1.16e14T^{2} \)
23 \( 1 - 4.13e7T + 9.52e14T^{2} \)
29 \( 1 + 1.20e8T + 1.22e16T^{2} \)
31 \( 1 + 2.94e8T + 2.54e16T^{2} \)
37 \( 1 + 2.60e8T + 1.77e17T^{2} \)
41 \( 1 + 1.02e9T + 5.50e17T^{2} \)
43 \( 1 - 8.13e8T + 9.29e17T^{2} \)
47 \( 1 + 1.49e9T + 2.47e18T^{2} \)
53 \( 1 + 1.26e9T + 9.26e18T^{2} \)
61 \( 1 + 1.28e10T + 4.35e19T^{2} \)
67 \( 1 + 1.20e10T + 1.22e20T^{2} \)
71 \( 1 - 2.83e10T + 2.31e20T^{2} \)
73 \( 1 + 1.47e10T + 3.13e20T^{2} \)
79 \( 1 - 1.46e10T + 7.47e20T^{2} \)
83 \( 1 + 2.54e10T + 1.28e21T^{2} \)
89 \( 1 + 3.02e10T + 2.77e21T^{2} \)
97 \( 1 - 6.30e10T + 7.15e21T^{2} \)
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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

−10.85793200761148520364185138062, −9.360937599142508517351740511239, −7.39108583626961890492711076834, −6.77511492300153603212075583522, −5.79552427349760825655546237071, −4.89651536209279265273541380585, −3.76387829308479475545669338142, −3.13688422753022098929748091222, −1.67783246404550662873027400291, 0, 1.67783246404550662873027400291, 3.13688422753022098929748091222, 3.76387829308479475545669338142, 4.89651536209279265273541380585, 5.79552427349760825655546237071, 6.77511492300153603212075583522, 7.39108583626961890492711076834, 9.360937599142508517351740511239, 10.85793200761148520364185138062

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