Properties

Label 2-177-1.1-c9-0-46
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.53·2-s + 81·3-s − 509.·4-s + 521.·5-s − 124.·6-s + 1.13e4·7-s + 1.56e3·8-s + 6.56e3·9-s − 800.·10-s + 9.51e4·11-s − 4.12e4·12-s + 5.89e4·13-s − 1.74e4·14-s + 4.22e4·15-s + 2.58e5·16-s − 8.92e4·17-s − 1.00e4·18-s + 3.08e5·19-s − 2.65e5·20-s + 9.19e5·21-s − 1.46e5·22-s + 2.23e6·23-s + 1.27e5·24-s − 1.68e6·25-s − 9.04e4·26-s + 5.31e5·27-s − 5.78e6·28-s + ⋯
L(s)  = 1  − 0.0678·2-s + 0.577·3-s − 0.995·4-s + 0.373·5-s − 0.0391·6-s + 1.78·7-s + 0.135·8-s + 0.333·9-s − 0.0253·10-s + 1.95·11-s − 0.574·12-s + 0.572·13-s − 0.121·14-s + 0.215·15-s + 0.986·16-s − 0.259·17-s − 0.0226·18-s + 0.543·19-s − 0.371·20-s + 1.03·21-s − 0.132·22-s + 1.66·23-s + 0.0781·24-s − 0.860·25-s − 0.0388·26-s + 0.192·27-s − 1.77·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(3.635091070\)
\(L(\frac12)\) \(\approx\) \(3.635091070\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 81T \)
59 \( 1 + 1.21e7T \)
good2 \( 1 + 1.53T + 512T^{2} \)
5 \( 1 - 521.T + 1.95e6T^{2} \)
7 \( 1 - 1.13e4T + 4.03e7T^{2} \)
11 \( 1 - 9.51e4T + 2.35e9T^{2} \)
13 \( 1 - 5.89e4T + 1.06e10T^{2} \)
17 \( 1 + 8.92e4T + 1.18e11T^{2} \)
19 \( 1 - 3.08e5T + 3.22e11T^{2} \)
23 \( 1 - 2.23e6T + 1.80e12T^{2} \)
29 \( 1 + 7.46e6T + 1.45e13T^{2} \)
31 \( 1 - 6.83e6T + 2.64e13T^{2} \)
37 \( 1 + 1.22e7T + 1.29e14T^{2} \)
41 \( 1 - 5.99e6T + 3.27e14T^{2} \)
43 \( 1 + 2.75e7T + 5.02e14T^{2} \)
47 \( 1 + 1.84e6T + 1.11e15T^{2} \)
53 \( 1 - 2.55e7T + 3.29e15T^{2} \)
61 \( 1 - 7.45e7T + 1.16e16T^{2} \)
67 \( 1 - 7.95e5T + 2.72e16T^{2} \)
71 \( 1 - 2.30e8T + 4.58e16T^{2} \)
73 \( 1 - 5.28e7T + 5.88e16T^{2} \)
79 \( 1 + 4.07e8T + 1.19e17T^{2} \)
83 \( 1 + 6.59e8T + 1.86e17T^{2} \)
89 \( 1 + 5.89e8T + 3.50e17T^{2} \)
97 \( 1 - 1.40e9T + 7.60e17T^{2} \)
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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

−11.10401839714127661033357752229, −9.688811202943410782771380625916, −8.911914015617596895023751991555, −8.292462626955461413653695809205, −7.07825514115617031436021520560, −5.52485681505150781876834174626, −4.48730663816803498285077990789, −3.61799416615980755237759769659, −1.70340853377238033158432057115, −1.08958825660705543096588106000, 1.08958825660705543096588106000, 1.70340853377238033158432057115, 3.61799416615980755237759769659, 4.48730663816803498285077990789, 5.52485681505150781876834174626, 7.07825514115617031436021520560, 8.292462626955461413653695809205, 8.911914015617596895023751991555, 9.688811202943410782771380625916, 11.10401839714127661033357752229

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