Properties

Label 2-177-1.1-c9-0-72
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  + 43.3·2-s + 81·3-s + 1.36e3·4-s + 2.36e3·5-s + 3.51e3·6-s − 2.16e3·7-s + 3.71e4·8-s + 6.56e3·9-s + 1.02e5·10-s − 4.65e4·11-s + 1.10e5·12-s − 2.93e4·13-s − 9.37e4·14-s + 1.91e5·15-s + 9.10e5·16-s − 3.36e5·17-s + 2.84e5·18-s + 6.88e5·19-s + 3.23e6·20-s − 1.75e5·21-s − 2.01e6·22-s + 1.45e6·23-s + 3.01e6·24-s + 3.62e6·25-s − 1.27e6·26-s + 5.31e5·27-s − 2.95e6·28-s + ⋯
L(s)  = 1  + 1.91·2-s + 0.577·3-s + 2.67·4-s + 1.69·5-s + 1.10·6-s − 0.340·7-s + 3.20·8-s + 0.333·9-s + 3.23·10-s − 0.958·11-s + 1.54·12-s − 0.285·13-s − 0.652·14-s + 0.975·15-s + 3.47·16-s − 0.976·17-s + 0.638·18-s + 1.21·19-s + 4.51·20-s − 0.196·21-s − 1.83·22-s + 1.08·23-s + 1.85·24-s + 1.85·25-s − 0.546·26-s + 0.192·27-s − 0.909·28-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

\(L(5)\) \(\approx\) \(12.85901940\)
\(L(\frac12)\) \(\approx\) \(12.85901940\)
\(L(\frac{11}{2})\) 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 \( 1 - 43.3T + 512T^{2} \)
5 \( 1 - 2.36e3T + 1.95e6T^{2} \)
7 \( 1 + 2.16e3T + 4.03e7T^{2} \)
11 \( 1 + 4.65e4T + 2.35e9T^{2} \)
13 \( 1 + 2.93e4T + 1.06e10T^{2} \)
17 \( 1 + 3.36e5T + 1.18e11T^{2} \)
19 \( 1 - 6.88e5T + 3.22e11T^{2} \)
23 \( 1 - 1.45e6T + 1.80e12T^{2} \)
29 \( 1 - 2.52e6T + 1.45e13T^{2} \)
31 \( 1 + 7.65e6T + 2.64e13T^{2} \)
37 \( 1 + 1.29e7T + 1.29e14T^{2} \)
41 \( 1 - 3.92e6T + 3.27e14T^{2} \)
43 \( 1 - 1.67e7T + 5.02e14T^{2} \)
47 \( 1 - 1.71e7T + 1.11e15T^{2} \)
53 \( 1 + 1.13e8T + 3.29e15T^{2} \)
61 \( 1 + 2.96e7T + 1.16e16T^{2} \)
67 \( 1 + 6.00e7T + 2.72e16T^{2} \)
71 \( 1 - 8.90e7T + 4.58e16T^{2} \)
73 \( 1 - 2.68e8T + 5.88e16T^{2} \)
79 \( 1 + 4.37e8T + 1.19e17T^{2} \)
83 \( 1 + 4.43e8T + 1.86e17T^{2} \)
89 \( 1 - 3.59e8T + 3.50e17T^{2} \)
97 \( 1 + 6.74e8T + 7.60e17T^{2} \)
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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

−11.13586736648626718953120198215, −10.27150297380405423558013447731, −9.196540850994065904147432694578, −7.41101070861927250623736837025, −6.50588481112312010775444003420, −5.48959488026216035458439407362, −4.82489507174976390440764099350, −3.22620868517848746252749113709, −2.51555904440685158269258858874, −1.60092418541279185467575921379, 1.60092418541279185467575921379, 2.51555904440685158269258858874, 3.22620868517848746252749113709, 4.82489507174976390440764099350, 5.48959488026216035458439407362, 6.50588481112312010775444003420, 7.41101070861927250623736837025, 9.196540850994065904147432694578, 10.27150297380405423558013447731, 11.13586736648626718953120198215

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