Properties

Label 2-177-1.1-c11-0-68
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 63.4·2-s + 243·3-s + 1.98e3·4-s − 1.37e3·5-s − 1.54e4·6-s − 1.61e4·7-s + 4.19e3·8-s + 5.90e4·9-s + 8.72e4·10-s + 3.91e5·11-s + 4.81e5·12-s − 1.24e6·13-s + 1.02e6·14-s − 3.33e5·15-s − 4.32e6·16-s + 8.24e5·17-s − 3.74e6·18-s + 4.80e6·19-s − 2.72e6·20-s − 3.91e6·21-s − 2.48e7·22-s + 2.79e7·23-s + 1.01e6·24-s − 4.69e7·25-s + 7.88e7·26-s + 1.43e7·27-s − 3.19e7·28-s + ⋯
L(s)  = 1  − 1.40·2-s + 0.577·3-s + 0.967·4-s − 0.196·5-s − 0.809·6-s − 0.362·7-s + 0.0452·8-s + 0.333·9-s + 0.275·10-s + 0.733·11-s + 0.558·12-s − 0.928·13-s + 0.508·14-s − 0.113·15-s − 1.03·16-s + 0.140·17-s − 0.467·18-s + 0.444·19-s − 0.190·20-s − 0.209·21-s − 1.02·22-s + 0.905·23-s + 0.0261·24-s − 0.961·25-s + 1.30·26-s + 0.192·27-s − 0.350·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) 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 \( 1 + 63.4T + 2.04e3T^{2} \)
5 \( 1 + 1.37e3T + 4.88e7T^{2} \)
7 \( 1 + 1.61e4T + 1.97e9T^{2} \)
11 \( 1 - 3.91e5T + 2.85e11T^{2} \)
13 \( 1 + 1.24e6T + 1.79e12T^{2} \)
17 \( 1 - 8.24e5T + 3.42e13T^{2} \)
19 \( 1 - 4.80e6T + 1.16e14T^{2} \)
23 \( 1 - 2.79e7T + 9.52e14T^{2} \)
29 \( 1 - 4.28e7T + 1.22e16T^{2} \)
31 \( 1 + 1.46e8T + 2.54e16T^{2} \)
37 \( 1 - 6.91e8T + 1.77e17T^{2} \)
41 \( 1 + 1.20e9T + 5.50e17T^{2} \)
43 \( 1 - 3.82e8T + 9.29e17T^{2} \)
47 \( 1 + 1.39e7T + 2.47e18T^{2} \)
53 \( 1 + 4.64e9T + 9.26e18T^{2} \)
61 \( 1 + 4.97e9T + 4.35e19T^{2} \)
67 \( 1 - 1.85e10T + 1.22e20T^{2} \)
71 \( 1 - 4.49e9T + 2.31e20T^{2} \)
73 \( 1 - 2.44e10T + 3.13e20T^{2} \)
79 \( 1 + 1.48e10T + 7.47e20T^{2} \)
83 \( 1 - 5.49e10T + 1.28e21T^{2} \)
89 \( 1 - 4.35e10T + 2.77e21T^{2} \)
97 \( 1 - 1.25e11T + 7.15e21T^{2} \)
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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

−9.610589458472818937517459641461, −9.436025212924211326034899760749, −8.222148266764613421070165542288, −7.47109154949468466170413839213, −6.54232579890647783852072073266, −4.80736835800237871170973400621, −3.46520285443221386886267387572, −2.19538910436718117894083529947, −1.10941315624682465188586038376, 0, 1.10941315624682465188586038376, 2.19538910436718117894083529947, 3.46520285443221386886267387572, 4.80736835800237871170973400621, 6.54232579890647783852072073266, 7.47109154949468466170413839213, 8.222148266764613421070165542288, 9.436025212924211326034899760749, 9.610589458472818937517459641461

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