Properties

Label 2-177-1.1-c13-0-73
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $189.798$
Root an. cond. $13.7767$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 73.5·2-s − 729·3-s − 2.78e3·4-s − 4.32e4·5-s − 5.35e4·6-s + 4.64e5·7-s − 8.07e5·8-s + 5.31e5·9-s − 3.18e6·10-s + 8.32e5·11-s + 2.03e6·12-s − 2.15e7·13-s + 3.41e7·14-s + 3.15e7·15-s − 3.65e7·16-s − 2.26e6·17-s + 3.90e7·18-s − 2.04e7·19-s + 1.20e8·20-s − 3.38e8·21-s + 6.11e7·22-s + 6.22e7·23-s + 5.88e8·24-s + 6.51e8·25-s − 1.58e9·26-s − 3.87e8·27-s − 1.29e9·28-s + ⋯
L(s)  = 1  + 0.812·2-s − 0.577·3-s − 0.340·4-s − 1.23·5-s − 0.468·6-s + 1.49·7-s − 1.08·8-s + 0.333·9-s − 1.00·10-s + 0.141·11-s + 0.196·12-s − 1.23·13-s + 1.21·14-s + 0.715·15-s − 0.543·16-s − 0.0227·17-s + 0.270·18-s − 0.0996·19-s + 0.421·20-s − 0.860·21-s + 0.115·22-s + 0.0876·23-s + 0.628·24-s + 0.533·25-s − 1.00·26-s − 0.192·27-s − 0.507·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(189.798\)
Root analytic conductor: \(13.7767\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 729T \)
59 \( 1 + 4.21e10T \)
good2 \( 1 - 73.5T + 8.19e3T^{2} \)
5 \( 1 + 4.32e4T + 1.22e9T^{2} \)
7 \( 1 - 4.64e5T + 9.68e10T^{2} \)
11 \( 1 - 8.32e5T + 3.45e13T^{2} \)
13 \( 1 + 2.15e7T + 3.02e14T^{2} \)
17 \( 1 + 2.26e6T + 9.90e15T^{2} \)
19 \( 1 + 2.04e7T + 4.20e16T^{2} \)
23 \( 1 - 6.22e7T + 5.04e17T^{2} \)
29 \( 1 - 4.72e9T + 1.02e19T^{2} \)
31 \( 1 - 8.27e9T + 2.44e19T^{2} \)
37 \( 1 + 2.05e10T + 2.43e20T^{2} \)
41 \( 1 - 1.73e10T + 9.25e20T^{2} \)
43 \( 1 - 2.60e10T + 1.71e21T^{2} \)
47 \( 1 + 7.14e10T + 5.46e21T^{2} \)
53 \( 1 - 1.22e11T + 2.60e22T^{2} \)
61 \( 1 - 5.55e11T + 1.61e23T^{2} \)
67 \( 1 - 6.72e11T + 5.48e23T^{2} \)
71 \( 1 + 1.20e12T + 1.16e24T^{2} \)
73 \( 1 - 6.36e11T + 1.67e24T^{2} \)
79 \( 1 + 1.84e12T + 4.66e24T^{2} \)
83 \( 1 - 5.07e12T + 8.87e24T^{2} \)
89 \( 1 + 3.61e12T + 2.19e25T^{2} \)
97 \( 1 + 6.16e12T + 6.73e25T^{2} \)
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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

−10.03324000999517281549347824067, −8.574481979034275892530558470754, −7.86054464854222376894044372184, −6.72430315203323716603565711191, −5.25658302993713503286150498847, −4.67553778537806614275493134199, −4.00471625671668656533612868553, −2.61138292921535480936284818552, −1.00846060260983124737975779465, 0, 1.00846060260983124737975779465, 2.61138292921535480936284818552, 4.00471625671668656533612868553, 4.67553778537806614275493134199, 5.25658302993713503286150498847, 6.72430315203323716603565711191, 7.86054464854222376894044372184, 8.574481979034275892530558470754, 10.03324000999517281549347824067

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