Properties

Label 2-177-1.1-c11-0-79
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  + 33.0·2-s + 243·3-s − 957.·4-s − 9.84e3·5-s + 8.02e3·6-s + 6.60e4·7-s − 9.92e4·8-s + 5.90e4·9-s − 3.24e5·10-s − 9.49e5·11-s − 2.32e5·12-s + 2.41e6·13-s + 2.17e6·14-s − 2.39e6·15-s − 1.31e6·16-s + 3.48e6·17-s + 1.94e6·18-s + 4.12e6·19-s + 9.42e6·20-s + 1.60e7·21-s − 3.13e7·22-s + 6.06e6·23-s − 2.41e7·24-s + 4.80e7·25-s + 7.96e7·26-s + 1.43e7·27-s − 6.32e7·28-s + ⋯
L(s)  = 1  + 0.729·2-s + 0.577·3-s − 0.467·4-s − 1.40·5-s + 0.421·6-s + 1.48·7-s − 1.07·8-s + 0.333·9-s − 1.02·10-s − 1.77·11-s − 0.269·12-s + 1.80·13-s + 1.08·14-s − 0.813·15-s − 0.313·16-s + 0.595·17-s + 0.243·18-s + 0.381·19-s + 0.658·20-s + 0.857·21-s − 1.29·22-s + 0.196·23-s − 0.618·24-s + 0.983·25-s + 1.31·26-s + 0.192·27-s − 0.694·28-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 \( 1 - 33.0T + 2.04e3T^{2} \)
5 \( 1 + 9.84e3T + 4.88e7T^{2} \)
7 \( 1 - 6.60e4T + 1.97e9T^{2} \)
11 \( 1 + 9.49e5T + 2.85e11T^{2} \)
13 \( 1 - 2.41e6T + 1.79e12T^{2} \)
17 \( 1 - 3.48e6T + 3.42e13T^{2} \)
19 \( 1 - 4.12e6T + 1.16e14T^{2} \)
23 \( 1 - 6.06e6T + 9.52e14T^{2} \)
29 \( 1 - 2.69e7T + 1.22e16T^{2} \)
31 \( 1 + 5.69e6T + 2.54e16T^{2} \)
37 \( 1 + 5.82e8T + 1.77e17T^{2} \)
41 \( 1 + 8.77e8T + 5.50e17T^{2} \)
43 \( 1 - 1.13e9T + 9.29e17T^{2} \)
47 \( 1 + 2.51e9T + 2.47e18T^{2} \)
53 \( 1 - 4.99e9T + 9.26e18T^{2} \)
61 \( 1 + 2.72e9T + 4.35e19T^{2} \)
67 \( 1 + 1.65e10T + 1.22e20T^{2} \)
71 \( 1 + 2.39e10T + 2.31e20T^{2} \)
73 \( 1 - 2.31e10T + 3.13e20T^{2} \)
79 \( 1 + 4.76e10T + 7.47e20T^{2} \)
83 \( 1 + 3.98e10T + 1.28e21T^{2} \)
89 \( 1 - 3.08e10T + 2.77e21T^{2} \)
97 \( 1 + 8.23e10T + 7.15e21T^{2} \)
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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

−10.41531889151699702052023906635, −8.597313773493366494384975089406, −8.286966823840981358980624796332, −7.43559107334063965519567511073, −5.56557887638284668778137360564, −4.73217220293984252123444473662, −3.80959696687344374970682254840, −2.98352421610975236494133123191, −1.30713781877628312156761689274, 0, 1.30713781877628312156761689274, 2.98352421610975236494133123191, 3.80959696687344374970682254840, 4.73217220293984252123444473662, 5.56557887638284668778137360564, 7.43559107334063965519567511073, 8.286966823840981358980624796332, 8.597313773493366494384975089406, 10.41531889151699702052023906635

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