Properties

Label 2-177-1.1-c13-0-31
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $189.798$
Root an. cond. $13.7767$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 118.·2-s + 729·3-s + 5.76e3·4-s + 218.·5-s − 8.61e4·6-s + 2.17e5·7-s + 2.86e5·8-s + 5.31e5·9-s − 2.57e4·10-s + 1.28e6·11-s + 4.20e6·12-s − 2.15e7·13-s − 2.57e7·14-s + 1.59e5·15-s − 8.10e7·16-s − 7.11e7·17-s − 6.27e7·18-s + 2.24e8·19-s + 1.25e6·20-s + 1.58e8·21-s − 1.51e8·22-s − 4.97e8·23-s + 2.08e8·24-s − 1.22e9·25-s + 2.54e9·26-s + 3.87e8·27-s + 1.25e9·28-s + ⋯
L(s)  = 1  − 1.30·2-s + 0.577·3-s + 0.704·4-s + 0.00624·5-s − 0.753·6-s + 0.699·7-s + 0.386·8-s + 0.333·9-s − 0.00815·10-s + 0.218·11-s + 0.406·12-s − 1.23·13-s − 0.912·14-s + 0.00360·15-s − 1.20·16-s − 0.714·17-s − 0.435·18-s + 1.09·19-s + 0.00439·20-s + 0.403·21-s − 0.285·22-s − 0.700·23-s + 0.222·24-s − 0.999·25-s + 1.61·26-s + 0.192·27-s + 0.492·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(189.798\)
Root analytic conductor: \(13.7767\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(1.197826109\)
\(L(\frac12)\) \(\approx\) \(1.197826109\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 729T \)
59 \( 1 + 4.21e10T \)
good2 \( 1 + 118.T + 8.19e3T^{2} \)
5 \( 1 - 218.T + 1.22e9T^{2} \)
7 \( 1 - 2.17e5T + 9.68e10T^{2} \)
11 \( 1 - 1.28e6T + 3.45e13T^{2} \)
13 \( 1 + 2.15e7T + 3.02e14T^{2} \)
17 \( 1 + 7.11e7T + 9.90e15T^{2} \)
19 \( 1 - 2.24e8T + 4.20e16T^{2} \)
23 \( 1 + 4.97e8T + 5.04e17T^{2} \)
29 \( 1 + 9.40e8T + 1.02e19T^{2} \)
31 \( 1 - 1.59e9T + 2.44e19T^{2} \)
37 \( 1 - 2.40e10T + 2.43e20T^{2} \)
41 \( 1 - 7.98e9T + 9.25e20T^{2} \)
43 \( 1 + 5.33e10T + 1.71e21T^{2} \)
47 \( 1 + 2.52e10T + 5.46e21T^{2} \)
53 \( 1 - 1.07e11T + 2.60e22T^{2} \)
61 \( 1 - 4.38e11T + 1.61e23T^{2} \)
67 \( 1 - 1.41e12T + 5.48e23T^{2} \)
71 \( 1 - 1.03e12T + 1.16e24T^{2} \)
73 \( 1 + 1.02e12T + 1.67e24T^{2} \)
79 \( 1 - 2.77e12T + 4.66e24T^{2} \)
83 \( 1 - 3.08e12T + 8.87e24T^{2} \)
89 \( 1 + 3.67e12T + 2.19e25T^{2} \)
97 \( 1 + 1.37e13T + 6.73e25T^{2} \)
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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

−9.812120128523477899044061216861, −9.530373464206310769495870712503, −8.265276374360382138847215755713, −7.77531065770880297463135575183, −6.79789617992087497530259470056, −5.12454775977037812412649505447, −4.06968061709756375066372337956, −2.47554259889830689646378003836, −1.68070781555877566934321533656, −0.55672295663972193912137231918, 0.55672295663972193912137231918, 1.68070781555877566934321533656, 2.47554259889830689646378003836, 4.06968061709756375066372337956, 5.12454775977037812412649505447, 6.79789617992087497530259470056, 7.77531065770880297463135575183, 8.265276374360382138847215755713, 9.530373464206310769495870712503, 9.812120128523477899044061216861

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