Properties

Label 2-177-1.1-c11-0-77
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  + 69.6·2-s − 243·3-s + 2.80e3·4-s + 1.29e4·5-s − 1.69e4·6-s + 4.20e4·7-s + 5.30e4·8-s + 5.90e4·9-s + 9.03e5·10-s + 7.08e5·11-s − 6.82e5·12-s + 4.20e5·13-s + 2.93e6·14-s − 3.15e6·15-s − 2.05e6·16-s − 2.22e6·17-s + 4.11e6·18-s + 9.57e6·19-s + 3.64e7·20-s − 1.02e7·21-s + 4.93e7·22-s + 2.54e7·23-s − 1.28e7·24-s + 1.19e8·25-s + 2.93e7·26-s − 1.43e7·27-s + 1.18e8·28-s + ⋯
L(s)  = 1  + 1.53·2-s − 0.577·3-s + 1.37·4-s + 1.85·5-s − 0.889·6-s + 0.946·7-s + 0.572·8-s + 0.333·9-s + 2.85·10-s + 1.32·11-s − 0.791·12-s + 0.314·13-s + 1.45·14-s − 1.07·15-s − 0.490·16-s − 0.380·17-s + 0.513·18-s + 0.886·19-s + 2.54·20-s − 0.546·21-s + 2.04·22-s + 0.823·23-s − 0.330·24-s + 2.44·25-s + 0.484·26-s − 0.192·27-s + 1.29·28-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

\(L(6)\) \(\approx\) \(8.707019756\)
\(L(\frac12)\) \(\approx\) \(8.707019756\)
\(L(\frac{13}{2})\) 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 \( 1 - 69.6T + 2.04e3T^{2} \)
5 \( 1 - 1.29e4T + 4.88e7T^{2} \)
7 \( 1 - 4.20e4T + 1.97e9T^{2} \)
11 \( 1 - 7.08e5T + 2.85e11T^{2} \)
13 \( 1 - 4.20e5T + 1.79e12T^{2} \)
17 \( 1 + 2.22e6T + 3.42e13T^{2} \)
19 \( 1 - 9.57e6T + 1.16e14T^{2} \)
23 \( 1 - 2.54e7T + 9.52e14T^{2} \)
29 \( 1 - 5.34e7T + 1.22e16T^{2} \)
31 \( 1 + 1.36e8T + 2.54e16T^{2} \)
37 \( 1 + 4.09e7T + 1.77e17T^{2} \)
41 \( 1 + 2.86e8T + 5.50e17T^{2} \)
43 \( 1 + 8.43e8T + 9.29e17T^{2} \)
47 \( 1 + 2.70e9T + 2.47e18T^{2} \)
53 \( 1 + 2.68e9T + 9.26e18T^{2} \)
61 \( 1 - 9.12e9T + 4.35e19T^{2} \)
67 \( 1 - 6.58e9T + 1.22e20T^{2} \)
71 \( 1 - 4.73e9T + 2.31e20T^{2} \)
73 \( 1 + 1.11e10T + 3.13e20T^{2} \)
79 \( 1 + 3.52e10T + 7.47e20T^{2} \)
83 \( 1 - 7.57e8T + 1.28e21T^{2} \)
89 \( 1 - 7.20e10T + 2.77e21T^{2} \)
97 \( 1 - 1.04e11T + 7.15e21T^{2} \)
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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

−11.08378620270806516315560739792, −9.814410963245017965359966411110, −8.853429871932717548299507980166, −6.86411871381210927300315116051, −6.23488889640830395303410867013, −5.30027092199689169413567809166, −4.73559957705942895389695832110, −3.32814807251486210615161871519, −1.95146849492535439345364545358, −1.26264593660044796689635475523, 1.26264593660044796689635475523, 1.95146849492535439345364545358, 3.32814807251486210615161871519, 4.73559957705942895389695832110, 5.30027092199689169413567809166, 6.23488889640830395303410867013, 6.86411871381210927300315116051, 8.853429871932717548299507980166, 9.814410963245017965359966411110, 11.08378620270806516315560739792

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