Properties

Label 2-177-1.1-c13-0-74
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 174.·2-s + 729·3-s + 2.23e4·4-s − 3.85e4·5-s + 1.27e5·6-s − 2.36e5·7-s + 2.47e6·8-s + 5.31e5·9-s − 6.73e6·10-s + 7.91e6·11-s + 1.62e7·12-s + 3.86e6·13-s − 4.12e7·14-s − 2.81e7·15-s + 2.49e8·16-s − 8.38e7·17-s + 9.28e7·18-s + 1.09e8·19-s − 8.61e8·20-s − 1.72e8·21-s + 1.38e9·22-s + 5.43e8·23-s + 1.80e9·24-s + 2.66e8·25-s + 6.75e8·26-s + 3.87e8·27-s − 5.27e9·28-s + ⋯
L(s)  = 1  + 1.93·2-s + 0.577·3-s + 2.72·4-s − 1.10·5-s + 1.11·6-s − 0.758·7-s + 3.33·8-s + 0.333·9-s − 2.13·10-s + 1.34·11-s + 1.57·12-s + 0.222·13-s − 1.46·14-s − 0.637·15-s + 3.71·16-s − 0.842·17-s + 0.643·18-s + 0.535·19-s − 3.01·20-s − 0.437·21-s + 2.59·22-s + 0.765·23-s + 1.92·24-s + 0.218·25-s + 0.428·26-s + 0.192·27-s − 2.06·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(10.10255946\)
\(L(\frac12)\) \(\approx\) \(10.10255946\)
\(L(\frac{15}{2})\) 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 \( 1 - 174.T + 8.19e3T^{2} \)
5 \( 1 + 3.85e4T + 1.22e9T^{2} \)
7 \( 1 + 2.36e5T + 9.68e10T^{2} \)
11 \( 1 - 7.91e6T + 3.45e13T^{2} \)
13 \( 1 - 3.86e6T + 3.02e14T^{2} \)
17 \( 1 + 8.38e7T + 9.90e15T^{2} \)
19 \( 1 - 1.09e8T + 4.20e16T^{2} \)
23 \( 1 - 5.43e8T + 5.04e17T^{2} \)
29 \( 1 + 6.42e8T + 1.02e19T^{2} \)
31 \( 1 - 6.97e9T + 2.44e19T^{2} \)
37 \( 1 + 2.07e9T + 2.43e20T^{2} \)
41 \( 1 - 2.58e10T + 9.25e20T^{2} \)
43 \( 1 + 5.08e9T + 1.71e21T^{2} \)
47 \( 1 - 5.45e10T + 5.46e21T^{2} \)
53 \( 1 + 8.09e10T + 2.60e22T^{2} \)
61 \( 1 - 8.52e10T + 1.61e23T^{2} \)
67 \( 1 - 6.87e11T + 5.48e23T^{2} \)
71 \( 1 - 1.01e12T + 1.16e24T^{2} \)
73 \( 1 - 1.75e12T + 1.67e24T^{2} \)
79 \( 1 - 1.74e12T + 4.66e24T^{2} \)
83 \( 1 - 2.86e12T + 8.87e24T^{2} \)
89 \( 1 - 7.08e12T + 2.19e25T^{2} \)
97 \( 1 - 1.48e12T + 6.73e25T^{2} \)
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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

−10.87664228457222571408676394297, −9.372751444931284540988421735793, −7.957313189235232707444149213816, −6.89616478289069836162339689998, −6.32264210342629343333440497153, −4.83084000343980337324173575923, −3.91160997281127374968141324993, −3.44331924046032676377382667392, −2.40053383700012406537476344311, −1.01358434183940714279543365911, 1.01358434183940714279543365911, 2.40053383700012406537476344311, 3.44331924046032676377382667392, 3.91160997281127374968141324993, 4.83084000343980337324173575923, 6.32264210342629343333440497153, 6.89616478289069836162339689998, 7.957313189235232707444149213816, 9.372751444931284540988421735793, 10.87664228457222571408676394297

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