Properties

Label 2-177-1.1-c11-0-75
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 90.0·2-s + 243·3-s + 6.05e3·4-s − 4.74e3·5-s + 2.18e4·6-s + 2.12e4·7-s + 3.60e5·8-s + 5.90e4·9-s − 4.27e5·10-s − 5.88e5·11-s + 1.47e6·12-s + 6.21e5·13-s + 1.91e6·14-s − 1.15e6·15-s + 2.00e7·16-s + 3.27e6·17-s + 5.31e6·18-s + 1.04e7·19-s − 2.87e7·20-s + 5.16e6·21-s − 5.29e7·22-s − 1.01e7·23-s + 8.76e7·24-s − 2.63e7·25-s + 5.59e7·26-s + 1.43e7·27-s + 1.28e8·28-s + ⋯
L(s)  = 1  + 1.98·2-s + 0.577·3-s + 2.95·4-s − 0.678·5-s + 1.14·6-s + 0.478·7-s + 3.89·8-s + 0.333·9-s − 1.35·10-s − 1.10·11-s + 1.70·12-s + 0.464·13-s + 0.951·14-s − 0.391·15-s + 4.78·16-s + 0.560·17-s + 0.663·18-s + 0.969·19-s − 2.00·20-s + 0.276·21-s − 2.19·22-s − 0.328·23-s + 2.24·24-s − 0.539·25-s + 0.923·26-s + 0.192·27-s + 1.41·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(11.81459282\)
\(L(\frac12)\) \(\approx\) \(11.81459282\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 243T \)
59 \( 1 - 7.14e8T \)
good2 \( 1 - 90.0T + 2.04e3T^{2} \)
5 \( 1 + 4.74e3T + 4.88e7T^{2} \)
7 \( 1 - 2.12e4T + 1.97e9T^{2} \)
11 \( 1 + 5.88e5T + 2.85e11T^{2} \)
13 \( 1 - 6.21e5T + 1.79e12T^{2} \)
17 \( 1 - 3.27e6T + 3.42e13T^{2} \)
19 \( 1 - 1.04e7T + 1.16e14T^{2} \)
23 \( 1 + 1.01e7T + 9.52e14T^{2} \)
29 \( 1 - 1.61e8T + 1.22e16T^{2} \)
31 \( 1 + 2.14e7T + 2.54e16T^{2} \)
37 \( 1 - 5.53e8T + 1.77e17T^{2} \)
41 \( 1 - 8.75e8T + 5.50e17T^{2} \)
43 \( 1 - 1.09e8T + 9.29e17T^{2} \)
47 \( 1 + 1.69e9T + 2.47e18T^{2} \)
53 \( 1 + 6.43e8T + 9.26e18T^{2} \)
61 \( 1 + 4.14e9T + 4.35e19T^{2} \)
67 \( 1 - 5.36e9T + 1.22e20T^{2} \)
71 \( 1 + 1.79e10T + 2.31e20T^{2} \)
73 \( 1 + 9.32e9T + 3.13e20T^{2} \)
79 \( 1 - 2.29e9T + 7.47e20T^{2} \)
83 \( 1 - 4.59e10T + 1.28e21T^{2} \)
89 \( 1 - 3.18e10T + 2.77e21T^{2} \)
97 \( 1 + 9.75e10T + 7.15e21T^{2} \)
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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

−11.12319439281487599551313886602, −10.09470734231381061795310665471, −7.957247992902341271732276108107, −7.62415101488003988471604602677, −6.24279637448218927123373062966, −5.16651065456003227562538133146, −4.30456753768966342933968208704, −3.31271083244166502046013935235, −2.52604353135594794728926606607, −1.24277591746954512852897234463, 1.24277591746954512852897234463, 2.52604353135594794728926606607, 3.31271083244166502046013935235, 4.30456753768966342933968208704, 5.16651065456003227562538133146, 6.24279637448218927123373062966, 7.62415101488003988471604602677, 7.957247992902341271732276108107, 10.09470734231381061795310665471, 11.12319439281487599551313886602

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