Properties

Label 2-177-1.1-c13-0-106
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 96.2·2-s − 729·3-s + 1.07e3·4-s + 2.66e4·5-s − 7.01e4·6-s + 2.74e5·7-s − 6.85e5·8-s + 5.31e5·9-s + 2.56e6·10-s + 8.00e5·11-s − 7.82e5·12-s − 3.00e6·13-s + 2.64e7·14-s − 1.93e7·15-s − 7.47e7·16-s − 1.21e8·17-s + 5.11e7·18-s + 4.02e8·19-s + 2.85e7·20-s − 2.00e8·21-s + 7.70e7·22-s − 6.66e8·23-s + 4.99e8·24-s − 5.12e8·25-s − 2.88e8·26-s − 3.87e8·27-s + 2.95e8·28-s + ⋯
L(s)  = 1  + 1.06·2-s − 0.577·3-s + 0.131·4-s + 0.761·5-s − 0.614·6-s + 0.882·7-s − 0.924·8-s + 0.333·9-s + 0.810·10-s + 0.136·11-s − 0.0756·12-s − 0.172·13-s + 0.938·14-s − 0.439·15-s − 1.11·16-s − 1.21·17-s + 0.354·18-s + 1.96·19-s + 0.0998·20-s − 0.509·21-s + 0.144·22-s − 0.938·23-s + 0.533·24-s − 0.419·25-s − 0.183·26-s − 0.192·27-s + 0.115·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) 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 \( 1 - 96.2T + 8.19e3T^{2} \)
5 \( 1 - 2.66e4T + 1.22e9T^{2} \)
7 \( 1 - 2.74e5T + 9.68e10T^{2} \)
11 \( 1 - 8.00e5T + 3.45e13T^{2} \)
13 \( 1 + 3.00e6T + 3.02e14T^{2} \)
17 \( 1 + 1.21e8T + 9.90e15T^{2} \)
19 \( 1 - 4.02e8T + 4.20e16T^{2} \)
23 \( 1 + 6.66e8T + 5.04e17T^{2} \)
29 \( 1 - 4.58e8T + 1.02e19T^{2} \)
31 \( 1 + 2.88e9T + 2.44e19T^{2} \)
37 \( 1 - 1.84e9T + 2.43e20T^{2} \)
41 \( 1 + 1.94e10T + 9.25e20T^{2} \)
43 \( 1 - 4.71e10T + 1.71e21T^{2} \)
47 \( 1 - 4.88e10T + 5.46e21T^{2} \)
53 \( 1 + 1.01e11T + 2.60e22T^{2} \)
61 \( 1 + 4.55e11T + 1.61e23T^{2} \)
67 \( 1 - 6.52e11T + 5.48e23T^{2} \)
71 \( 1 - 1.59e12T + 1.16e24T^{2} \)
73 \( 1 + 1.70e12T + 1.67e24T^{2} \)
79 \( 1 + 8.10e11T + 4.66e24T^{2} \)
83 \( 1 + 1.15e12T + 8.87e24T^{2} \)
89 \( 1 - 2.23e12T + 2.19e25T^{2} \)
97 \( 1 + 9.65e12T + 6.73e25T^{2} \)
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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

−9.877250103112043040850999177467, −9.017229851569019903035422339563, −7.62200152165571635814165837881, −6.35853635082896537303872157571, −5.51642921700244537968372438701, −4.84200535841653086658012816622, −3.84799859340455635386296802422, −2.46907024749161627140156158027, −1.37622936819117273897365478523, 0, 1.37622936819117273897365478523, 2.46907024749161627140156158027, 3.84799859340455635386296802422, 4.84200535841653086658012816622, 5.51642921700244537968372438701, 6.35853635082896537303872157571, 7.62200152165571635814165837881, 9.017229851569019903035422339563, 9.877250103112043040850999177467

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