Properties

Label 2-177-1.1-c9-0-39
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  + 38.7·2-s − 81·3-s + 990.·4-s − 782.·5-s − 3.13e3·6-s + 8.97e3·7-s + 1.85e4·8-s + 6.56e3·9-s − 3.03e4·10-s + 2.31e4·11-s − 8.02e4·12-s + 5.51e4·13-s + 3.47e5·14-s + 6.33e4·15-s + 2.11e5·16-s − 6.44e5·17-s + 2.54e5·18-s + 7.72e5·19-s − 7.74e5·20-s − 7.26e5·21-s + 8.98e5·22-s + 1.30e6·23-s − 1.50e6·24-s − 1.34e6·25-s + 2.13e6·26-s − 5.31e5·27-s + 8.88e6·28-s + ⋯
L(s)  = 1  + 1.71·2-s − 0.577·3-s + 1.93·4-s − 0.559·5-s − 0.989·6-s + 1.41·7-s + 1.60·8-s + 0.333·9-s − 0.959·10-s + 0.477·11-s − 1.11·12-s + 0.535·13-s + 2.41·14-s + 0.323·15-s + 0.808·16-s − 1.87·17-s + 0.571·18-s + 1.35·19-s − 1.08·20-s − 0.815·21-s + 0.817·22-s + 0.972·23-s − 0.924·24-s − 0.686·25-s + 0.918·26-s − 0.192·27-s + 2.73·28-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

\(L(5)\) \(\approx\) \(6.163150346\)
\(L(\frac12)\) \(\approx\) \(6.163150346\)
\(L(\frac{11}{2})\) 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 \( 1 - 38.7T + 512T^{2} \)
5 \( 1 + 782.T + 1.95e6T^{2} \)
7 \( 1 - 8.97e3T + 4.03e7T^{2} \)
11 \( 1 - 2.31e4T + 2.35e9T^{2} \)
13 \( 1 - 5.51e4T + 1.06e10T^{2} \)
17 \( 1 + 6.44e5T + 1.18e11T^{2} \)
19 \( 1 - 7.72e5T + 3.22e11T^{2} \)
23 \( 1 - 1.30e6T + 1.80e12T^{2} \)
29 \( 1 - 2.54e6T + 1.45e13T^{2} \)
31 \( 1 - 3.89e6T + 2.64e13T^{2} \)
37 \( 1 + 1.24e7T + 1.29e14T^{2} \)
41 \( 1 - 3.06e7T + 3.27e14T^{2} \)
43 \( 1 - 2.82e7T + 5.02e14T^{2} \)
47 \( 1 + 2.18e7T + 1.11e15T^{2} \)
53 \( 1 - 1.06e8T + 3.29e15T^{2} \)
61 \( 1 - 1.31e8T + 1.16e16T^{2} \)
67 \( 1 - 2.37e8T + 2.72e16T^{2} \)
71 \( 1 + 3.67e8T + 4.58e16T^{2} \)
73 \( 1 - 1.69e8T + 5.88e16T^{2} \)
79 \( 1 + 3.64e7T + 1.19e17T^{2} \)
83 \( 1 + 4.16e8T + 1.86e17T^{2} \)
89 \( 1 - 1.10e9T + 3.50e17T^{2} \)
97 \( 1 - 7.85e8T + 7.60e17T^{2} \)
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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

−11.52828096589626713645425877932, −10.75208347981598056246256234562, −8.846144194320148441641605921091, −7.50738565761256208123345462738, −6.56757574608624760692597754172, −5.41392061892348145090249619073, −4.59896685278826607396765396200, −3.86935546175914162245616752663, −2.35749240224942325380476225283, −1.03883636991027128441322094510, 1.03883636991027128441322094510, 2.35749240224942325380476225283, 3.86935546175914162245616752663, 4.59896685278826607396765396200, 5.41392061892348145090249619073, 6.56757574608624760692597754172, 7.50738565761256208123345462738, 8.846144194320148441641605921091, 10.75208347981598056246256234562, 11.52828096589626713645425877932

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