Properties

Label 2-177-1.1-c11-0-103
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  + 65.6·2-s + 243·3-s + 2.26e3·4-s − 3.29e3·5-s + 1.59e4·6-s + 5.47e4·7-s + 1.44e4·8-s + 5.90e4·9-s − 2.16e5·10-s − 4.23e5·11-s + 5.50e5·12-s − 1.58e6·13-s + 3.59e6·14-s − 8.00e5·15-s − 3.69e6·16-s − 4.57e6·17-s + 3.87e6·18-s + 1.41e7·19-s − 7.46e6·20-s + 1.32e7·21-s − 2.78e7·22-s − 1.71e7·23-s + 3.50e6·24-s − 3.79e7·25-s − 1.04e8·26-s + 1.43e7·27-s + 1.24e8·28-s + ⋯
L(s)  = 1  + 1.45·2-s + 0.577·3-s + 1.10·4-s − 0.471·5-s + 0.838·6-s + 1.23·7-s + 0.155·8-s + 0.333·9-s − 0.684·10-s − 0.793·11-s + 0.639·12-s − 1.18·13-s + 1.78·14-s − 0.272·15-s − 0.881·16-s − 0.781·17-s + 0.483·18-s + 1.31·19-s − 0.521·20-s + 0.710·21-s − 1.15·22-s − 0.554·23-s + 0.0897·24-s − 0.777·25-s − 1.71·26-s + 0.192·27-s + 1.36·28-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 \( 1 - 65.6T + 2.04e3T^{2} \)
5 \( 1 + 3.29e3T + 4.88e7T^{2} \)
7 \( 1 - 5.47e4T + 1.97e9T^{2} \)
11 \( 1 + 4.23e5T + 2.85e11T^{2} \)
13 \( 1 + 1.58e6T + 1.79e12T^{2} \)
17 \( 1 + 4.57e6T + 3.42e13T^{2} \)
19 \( 1 - 1.41e7T + 1.16e14T^{2} \)
23 \( 1 + 1.71e7T + 9.52e14T^{2} \)
29 \( 1 - 5.00e7T + 1.22e16T^{2} \)
31 \( 1 + 1.79e8T + 2.54e16T^{2} \)
37 \( 1 + 5.12e8T + 1.77e17T^{2} \)
41 \( 1 - 1.65e8T + 5.50e17T^{2} \)
43 \( 1 + 8.33e8T + 9.29e17T^{2} \)
47 \( 1 - 2.33e9T + 2.47e18T^{2} \)
53 \( 1 - 8.94e8T + 9.26e18T^{2} \)
61 \( 1 - 7.14e9T + 4.35e19T^{2} \)
67 \( 1 + 1.42e10T + 1.22e20T^{2} \)
71 \( 1 + 2.36e10T + 2.31e20T^{2} \)
73 \( 1 + 1.92e10T + 3.13e20T^{2} \)
79 \( 1 - 2.75e10T + 7.47e20T^{2} \)
83 \( 1 - 3.14e10T + 1.28e21T^{2} \)
89 \( 1 + 4.56e10T + 2.77e21T^{2} \)
97 \( 1 - 1.05e11T + 7.15e21T^{2} \)
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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

−10.41561744922112426129163896265, −9.041479936534016045752534989897, −7.85170027932225502202008566280, −7.15689600857885999219129180829, −5.50442573819868559810160672459, −4.81927463630784014169559919563, −3.91446493101410325044712308277, −2.76383506646131365276678382274, −1.85574121300937972195570664559, 0, 1.85574121300937972195570664559, 2.76383506646131365276678382274, 3.91446493101410325044712308277, 4.81927463630784014169559919563, 5.50442573819868559810160672459, 7.15689600857885999219129180829, 7.85170027932225502202008566280, 9.041479936534016045752534989897, 10.41561744922112426129163896265

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