Properties

Label 2-177-1.1-c9-0-69
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  + 42.2·2-s + 81·3-s + 1.27e3·4-s − 60.0·5-s + 3.42e3·6-s + 8.82e3·7-s + 3.22e4·8-s + 6.56e3·9-s − 2.53e3·10-s + 1.62e4·11-s + 1.03e5·12-s + 1.23e5·13-s + 3.73e5·14-s − 4.86e3·15-s + 7.10e5·16-s − 1.65e5·17-s + 2.77e5·18-s − 6.92e5·19-s − 7.65e4·20-s + 7.14e5·21-s + 6.84e5·22-s − 3.87e5·23-s + 2.61e6·24-s − 1.94e6·25-s + 5.21e6·26-s + 5.31e5·27-s + 1.12e7·28-s + ⋯
L(s)  = 1  + 1.86·2-s + 0.577·3-s + 2.48·4-s − 0.0429·5-s + 1.07·6-s + 1.38·7-s + 2.78·8-s + 0.333·9-s − 0.0802·10-s + 0.333·11-s + 1.43·12-s + 1.19·13-s + 2.59·14-s − 0.0247·15-s + 2.70·16-s − 0.481·17-s + 0.622·18-s − 1.21·19-s − 0.106·20-s + 0.802·21-s + 0.623·22-s − 0.288·23-s + 1.60·24-s − 0.998·25-s + 2.23·26-s + 0.192·27-s + 3.45·28-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

\(L(5)\) \(\approx\) \(11.60042244\)
\(L(\frac12)\) \(\approx\) \(11.60042244\)
\(L(\frac{11}{2})\) 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 \( 1 - 42.2T + 512T^{2} \)
5 \( 1 + 60.0T + 1.95e6T^{2} \)
7 \( 1 - 8.82e3T + 4.03e7T^{2} \)
11 \( 1 - 1.62e4T + 2.35e9T^{2} \)
13 \( 1 - 1.23e5T + 1.06e10T^{2} \)
17 \( 1 + 1.65e5T + 1.18e11T^{2} \)
19 \( 1 + 6.92e5T + 3.22e11T^{2} \)
23 \( 1 + 3.87e5T + 1.80e12T^{2} \)
29 \( 1 + 1.66e6T + 1.45e13T^{2} \)
31 \( 1 + 1.05e6T + 2.64e13T^{2} \)
37 \( 1 - 5.42e6T + 1.29e14T^{2} \)
41 \( 1 + 2.48e7T + 3.27e14T^{2} \)
43 \( 1 - 1.30e7T + 5.02e14T^{2} \)
47 \( 1 - 2.22e7T + 1.11e15T^{2} \)
53 \( 1 - 9.05e7T + 3.29e15T^{2} \)
61 \( 1 - 5.78e7T + 1.16e16T^{2} \)
67 \( 1 + 2.31e8T + 2.72e16T^{2} \)
71 \( 1 - 1.74e7T + 4.58e16T^{2} \)
73 \( 1 - 1.14e8T + 5.88e16T^{2} \)
79 \( 1 + 1.84e8T + 1.19e17T^{2} \)
83 \( 1 - 5.49e8T + 1.86e17T^{2} \)
89 \( 1 - 1.64e8T + 3.50e17T^{2} \)
97 \( 1 + 1.37e9T + 7.60e17T^{2} \)
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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

−11.31829156753377835342469981424, −10.58652019674453704678117475511, −8.703133831659072834656613766027, −7.73542509393641340054247013667, −6.54701039956571401456590672372, −5.53090460029737518370946168683, −4.34375572180625058566327585038, −3.78672741388179527864140700066, −2.30839817299573472746727752598, −1.52886575134937182375786154549, 1.52886575134937182375786154549, 2.30839817299573472746727752598, 3.78672741388179527864140700066, 4.34375572180625058566327585038, 5.53090460029737518370946168683, 6.54701039956571401456590672372, 7.73542509393641340054247013667, 8.703133831659072834656613766027, 10.58652019674453704678117475511, 11.31829156753377835342469981424

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