Properties

Label 2-177-1.1-c11-0-59
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  + 13.5·2-s + 243·3-s − 1.86e3·4-s − 1.20e4·5-s + 3.28e3·6-s − 1.96e4·7-s − 5.29e4·8-s + 5.90e4·9-s − 1.62e5·10-s + 2.75e5·11-s − 4.53e5·12-s + 2.60e5·13-s − 2.66e5·14-s − 2.91e6·15-s + 3.10e6·16-s + 2.78e6·17-s + 7.98e5·18-s + 1.56e7·19-s + 2.23e7·20-s − 4.78e6·21-s + 3.72e6·22-s − 5.51e7·23-s − 1.28e7·24-s + 9.52e7·25-s + 3.52e6·26-s + 1.43e7·27-s + 3.67e7·28-s + ⋯
L(s)  = 1  + 0.298·2-s + 0.577·3-s − 0.910·4-s − 1.71·5-s + 0.172·6-s − 0.442·7-s − 0.570·8-s + 0.333·9-s − 0.513·10-s + 0.516·11-s − 0.525·12-s + 0.194·13-s − 0.132·14-s − 0.991·15-s + 0.740·16-s + 0.475·17-s + 0.0995·18-s + 1.45·19-s + 1.56·20-s − 0.255·21-s + 0.154·22-s − 1.78·23-s − 0.329·24-s + 1.95·25-s + 0.0581·26-s + 0.192·27-s + 0.403·28-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 \( 1 - 13.5T + 2.04e3T^{2} \)
5 \( 1 + 1.20e4T + 4.88e7T^{2} \)
7 \( 1 + 1.96e4T + 1.97e9T^{2} \)
11 \( 1 - 2.75e5T + 2.85e11T^{2} \)
13 \( 1 - 2.60e5T + 1.79e12T^{2} \)
17 \( 1 - 2.78e6T + 3.42e13T^{2} \)
19 \( 1 - 1.56e7T + 1.16e14T^{2} \)
23 \( 1 + 5.51e7T + 9.52e14T^{2} \)
29 \( 1 + 4.30e7T + 1.22e16T^{2} \)
31 \( 1 - 1.37e7T + 2.54e16T^{2} \)
37 \( 1 - 2.61e8T + 1.77e17T^{2} \)
41 \( 1 - 1.38e9T + 5.50e17T^{2} \)
43 \( 1 - 4.91e8T + 9.29e17T^{2} \)
47 \( 1 - 9.26e8T + 2.47e18T^{2} \)
53 \( 1 + 5.11e9T + 9.26e18T^{2} \)
61 \( 1 - 5.97e9T + 4.35e19T^{2} \)
67 \( 1 - 1.02e10T + 1.22e20T^{2} \)
71 \( 1 + 2.24e10T + 2.31e20T^{2} \)
73 \( 1 + 2.72e10T + 3.13e20T^{2} \)
79 \( 1 + 2.62e10T + 7.47e20T^{2} \)
83 \( 1 + 4.00e10T + 1.28e21T^{2} \)
89 \( 1 - 6.06e10T + 2.77e21T^{2} \)
97 \( 1 - 1.53e11T + 7.15e21T^{2} \)
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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

−9.929691361184226346293728794126, −9.094653044810240094045043001120, −8.047112556020453492285290991186, −7.47781253085466476324683260981, −5.91227162842972485422160720642, −4.44927058526158075481911258407, −3.80275281049955630981404692488, −3.05603240694597786332462198151, −1.02210316695003791867412328194, 0, 1.02210316695003791867412328194, 3.05603240694597786332462198151, 3.80275281049955630981404692488, 4.44927058526158075481911258407, 5.91227162842972485422160720642, 7.47781253085466476324683260981, 8.047112556020453492285290991186, 9.094653044810240094045043001120, 9.929691361184226346293728794126

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