Properties

Label 2-177-1.1-c9-0-40
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.7·2-s − 81·3-s − 265.·4-s − 2.48e3·5-s − 1.27e3·6-s + 1.63e3·7-s − 1.22e4·8-s + 6.56e3·9-s − 3.89e4·10-s + 7.19e4·11-s + 2.14e4·12-s + 2.09e4·13-s + 2.56e4·14-s + 2.00e5·15-s − 5.60e4·16-s − 1.00e5·17-s + 1.03e5·18-s + 1.25e5·19-s + 6.58e5·20-s − 1.32e5·21-s + 1.13e6·22-s + 7.59e5·23-s + 9.88e5·24-s + 4.20e6·25-s + 3.29e5·26-s − 5.31e5·27-s − 4.32e5·28-s + ⋯
L(s)  = 1  + 0.694·2-s − 0.577·3-s − 0.517·4-s − 1.77·5-s − 0.400·6-s + 0.256·7-s − 1.05·8-s + 0.333·9-s − 1.23·10-s + 1.48·11-s + 0.299·12-s + 0.203·13-s + 0.178·14-s + 1.02·15-s − 0.213·16-s − 0.291·17-s + 0.231·18-s + 0.221·19-s + 0.919·20-s − 0.148·21-s + 1.02·22-s + 0.566·23-s + 0.608·24-s + 2.15·25-s + 0.141·26-s − 0.192·27-s − 0.133·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 - 15.7T + 512T^{2} \)
5 \( 1 + 2.48e3T + 1.95e6T^{2} \)
7 \( 1 - 1.63e3T + 4.03e7T^{2} \)
11 \( 1 - 7.19e4T + 2.35e9T^{2} \)
13 \( 1 - 2.09e4T + 1.06e10T^{2} \)
17 \( 1 + 1.00e5T + 1.18e11T^{2} \)
19 \( 1 - 1.25e5T + 3.22e11T^{2} \)
23 \( 1 - 7.59e5T + 1.80e12T^{2} \)
29 \( 1 + 1.44e6T + 1.45e13T^{2} \)
31 \( 1 + 9.73e5T + 2.64e13T^{2} \)
37 \( 1 - 6.01e6T + 1.29e14T^{2} \)
41 \( 1 - 6.93e6T + 3.27e14T^{2} \)
43 \( 1 - 5.22e6T + 5.02e14T^{2} \)
47 \( 1 + 3.13e7T + 1.11e15T^{2} \)
53 \( 1 - 5.51e7T + 3.29e15T^{2} \)
61 \( 1 + 1.66e8T + 1.16e16T^{2} \)
67 \( 1 - 9.31e7T + 2.72e16T^{2} \)
71 \( 1 + 4.01e7T + 4.58e16T^{2} \)
73 \( 1 + 2.58e8T + 5.88e16T^{2} \)
79 \( 1 + 3.62e8T + 1.19e17T^{2} \)
83 \( 1 - 5.97e8T + 1.86e17T^{2} \)
89 \( 1 - 6.86e8T + 3.50e17T^{2} \)
97 \( 1 - 2.73e7T + 7.60e17T^{2} \)
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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

−11.01490373996782137582599016526, −9.363501018999949435785872452799, −8.463683092847462050368631876798, −7.31062124175911882934108003689, −6.22297272579935510924896696978, −4.82811651340075572923889713922, −4.12358521646290477821288169132, −3.34562288392091298879938537186, −1.05276356004009634246129934718, 0, 1.05276356004009634246129934718, 3.34562288392091298879938537186, 4.12358521646290477821288169132, 4.82811651340075572923889713922, 6.22297272579935510924896696978, 7.31062124175911882934108003689, 8.463683092847462050368631876798, 9.363501018999949435785872452799, 11.01490373996782137582599016526

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