Properties

Label 2-177-1.1-c9-0-22
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  − 16.3·2-s + 81·3-s − 244.·4-s − 1.40e3·5-s − 1.32e3·6-s + 3.36e3·7-s + 1.23e4·8-s + 6.56e3·9-s + 2.29e4·10-s + 3.11e4·11-s − 1.98e4·12-s + 1.81e5·13-s − 5.49e4·14-s − 1.13e5·15-s − 7.68e4·16-s + 5.60e5·17-s − 1.07e5·18-s − 3.92e5·19-s + 3.43e5·20-s + 2.72e5·21-s − 5.09e5·22-s − 2.10e6·23-s + 1.00e6·24-s + 1.22e4·25-s − 2.96e6·26-s + 5.31e5·27-s − 8.22e5·28-s + ⋯
L(s)  = 1  − 0.722·2-s + 0.577·3-s − 0.478·4-s − 1.00·5-s − 0.417·6-s + 0.529·7-s + 1.06·8-s + 0.333·9-s + 0.724·10-s + 0.641·11-s − 0.276·12-s + 1.76·13-s − 0.382·14-s − 0.579·15-s − 0.293·16-s + 1.62·17-s − 0.240·18-s − 0.690·19-s + 0.479·20-s + 0.305·21-s − 0.463·22-s − 1.57·23-s + 0.616·24-s + 0.00628·25-s − 1.27·26-s + 0.192·27-s − 0.253·28-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

\(L(5)\) \(\approx\) \(1.545490030\)
\(L(\frac12)\) \(\approx\) \(1.545490030\)
\(L(\frac{11}{2})\) 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 \( 1 + 16.3T + 512T^{2} \)
5 \( 1 + 1.40e3T + 1.95e6T^{2} \)
7 \( 1 - 3.36e3T + 4.03e7T^{2} \)
11 \( 1 - 3.11e4T + 2.35e9T^{2} \)
13 \( 1 - 1.81e5T + 1.06e10T^{2} \)
17 \( 1 - 5.60e5T + 1.18e11T^{2} \)
19 \( 1 + 3.92e5T + 3.22e11T^{2} \)
23 \( 1 + 2.10e6T + 1.80e12T^{2} \)
29 \( 1 + 1.57e6T + 1.45e13T^{2} \)
31 \( 1 + 2.88e6T + 2.64e13T^{2} \)
37 \( 1 + 2.17e6T + 1.29e14T^{2} \)
41 \( 1 - 1.94e7T + 3.27e14T^{2} \)
43 \( 1 - 1.36e7T + 5.02e14T^{2} \)
47 \( 1 - 1.56e7T + 1.11e15T^{2} \)
53 \( 1 - 7.05e7T + 3.29e15T^{2} \)
61 \( 1 - 1.38e8T + 1.16e16T^{2} \)
67 \( 1 - 2.09e8T + 2.72e16T^{2} \)
71 \( 1 + 1.48e8T + 4.58e16T^{2} \)
73 \( 1 + 2.48e8T + 5.88e16T^{2} \)
79 \( 1 + 5.22e8T + 1.19e17T^{2} \)
83 \( 1 - 6.25e8T + 1.86e17T^{2} \)
89 \( 1 + 6.80e8T + 3.50e17T^{2} \)
97 \( 1 - 9.35e7T + 7.60e17T^{2} \)
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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

−10.88124912677140296493380922029, −9.858528907290897964143765841820, −8.738548667979126565923307057015, −8.171715095053106404191552182978, −7.45415627988429628182644848578, −5.80780415324782674596064736692, −4.11546993146637039062786615892, −3.70900352720534765389515905978, −1.67441583422730549531735155272, −0.72033056180980200294330306541, 0.72033056180980200294330306541, 1.67441583422730549531735155272, 3.70900352720534765389515905978, 4.11546993146637039062786615892, 5.80780415324782674596064736692, 7.45415627988429628182644848578, 8.171715095053106404191552182978, 8.738548667979126565923307057015, 9.858528907290897964143765841820, 10.88124912677140296493380922029

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