Properties

Label 2-177-1.1-c11-0-102
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  + 81.8·2-s + 243·3-s + 4.64e3·4-s − 1.04e4·5-s + 1.98e4·6-s − 3.31e4·7-s + 2.12e5·8-s + 5.90e4·9-s − 8.54e5·10-s + 6.92e5·11-s + 1.12e6·12-s − 1.94e6·13-s − 2.71e6·14-s − 2.53e6·15-s + 7.90e6·16-s + 9.86e6·17-s + 4.83e6·18-s − 8.54e6·19-s − 4.85e7·20-s − 8.06e6·21-s + 5.67e7·22-s − 1.31e7·23-s + 5.17e7·24-s + 6.00e7·25-s − 1.58e8·26-s + 1.43e7·27-s − 1.54e8·28-s + ⋯
L(s)  = 1  + 1.80·2-s + 0.577·3-s + 2.27·4-s − 1.49·5-s + 1.04·6-s − 0.746·7-s + 2.29·8-s + 0.333·9-s − 2.70·10-s + 1.29·11-s + 1.31·12-s − 1.44·13-s − 1.34·14-s − 0.862·15-s + 1.88·16-s + 1.68·17-s + 0.602·18-s − 0.791·19-s − 3.39·20-s − 0.430·21-s + 2.34·22-s − 0.424·23-s + 1.32·24-s + 1.23·25-s − 2.62·26-s + 0.192·27-s − 1.69·28-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 \( 1 - 81.8T + 2.04e3T^{2} \)
5 \( 1 + 1.04e4T + 4.88e7T^{2} \)
7 \( 1 + 3.31e4T + 1.97e9T^{2} \)
11 \( 1 - 6.92e5T + 2.85e11T^{2} \)
13 \( 1 + 1.94e6T + 1.79e12T^{2} \)
17 \( 1 - 9.86e6T + 3.42e13T^{2} \)
19 \( 1 + 8.54e6T + 1.16e14T^{2} \)
23 \( 1 + 1.31e7T + 9.52e14T^{2} \)
29 \( 1 + 1.43e8T + 1.22e16T^{2} \)
31 \( 1 + 2.66e8T + 2.54e16T^{2} \)
37 \( 1 + 6.10e8T + 1.77e17T^{2} \)
41 \( 1 + 8.43e8T + 5.50e17T^{2} \)
43 \( 1 - 1.21e9T + 9.29e17T^{2} \)
47 \( 1 - 9.46e8T + 2.47e18T^{2} \)
53 \( 1 - 1.89e9T + 9.26e18T^{2} \)
61 \( 1 + 9.26e9T + 4.35e19T^{2} \)
67 \( 1 + 1.35e10T + 1.22e20T^{2} \)
71 \( 1 + 2.44e10T + 2.31e20T^{2} \)
73 \( 1 + 1.73e10T + 3.13e20T^{2} \)
79 \( 1 - 8.03e9T + 7.47e20T^{2} \)
83 \( 1 - 3.54e10T + 1.28e21T^{2} \)
89 \( 1 - 5.14e10T + 2.77e21T^{2} \)
97 \( 1 + 1.11e11T + 7.15e21T^{2} \)
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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

−10.49555590835467881132248739125, −9.135262607741950285900713058170, −7.53300390868595038973303151047, −7.12938016931803311932764946308, −5.80639215924175946561683908905, −4.53003676378456576405259443099, −3.66212336393471813353920385962, −3.25661953692079311206709582357, −1.82784342811883613989702407655, 0, 1.82784342811883613989702407655, 3.25661953692079311206709582357, 3.66212336393471813353920385962, 4.53003676378456576405259443099, 5.80639215924175946561683908905, 7.12938016931803311932764946308, 7.53300390868595038973303151047, 9.135262607741950285900713058170, 10.49555590835467881132248739125

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