Properties

Label 2-177-1.1-c13-0-67
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  − 98.0·2-s − 729·3-s + 1.41e3·4-s + 4.39e4·5-s + 7.14e4·6-s − 3.96e5·7-s + 6.64e5·8-s + 5.31e5·9-s − 4.30e6·10-s − 1.79e6·11-s − 1.03e6·12-s + 6.51e6·13-s + 3.88e7·14-s − 3.20e7·15-s − 7.66e7·16-s + 9.76e7·17-s − 5.20e7·18-s − 6.83e7·19-s + 6.21e7·20-s + 2.89e8·21-s + 1.76e8·22-s + 1.72e8·23-s − 4.84e8·24-s + 7.11e8·25-s − 6.38e8·26-s − 3.87e8·27-s − 5.61e8·28-s + ⋯
L(s)  = 1  − 1.08·2-s − 0.577·3-s + 0.172·4-s + 1.25·5-s + 0.625·6-s − 1.27·7-s + 0.895·8-s + 0.333·9-s − 1.36·10-s − 0.306·11-s − 0.0997·12-s + 0.374·13-s + 1.38·14-s − 0.726·15-s − 1.14·16-s + 0.981·17-s − 0.360·18-s − 0.333·19-s + 0.217·20-s + 0.735·21-s + 0.331·22-s + 0.242·23-s − 0.517·24-s + 0.582·25-s − 0.405·26-s − 0.192·27-s − 0.220·28-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 \( 1 + 98.0T + 8.19e3T^{2} \)
5 \( 1 - 4.39e4T + 1.22e9T^{2} \)
7 \( 1 + 3.96e5T + 9.68e10T^{2} \)
11 \( 1 + 1.79e6T + 3.45e13T^{2} \)
13 \( 1 - 6.51e6T + 3.02e14T^{2} \)
17 \( 1 - 9.76e7T + 9.90e15T^{2} \)
19 \( 1 + 6.83e7T + 4.20e16T^{2} \)
23 \( 1 - 1.72e8T + 5.04e17T^{2} \)
29 \( 1 + 5.74e8T + 1.02e19T^{2} \)
31 \( 1 - 3.46e8T + 2.44e19T^{2} \)
37 \( 1 - 2.87e10T + 2.43e20T^{2} \)
41 \( 1 + 3.40e10T + 9.25e20T^{2} \)
43 \( 1 + 5.78e10T + 1.71e21T^{2} \)
47 \( 1 + 1.12e11T + 5.46e21T^{2} \)
53 \( 1 + 1.23e11T + 2.60e22T^{2} \)
61 \( 1 - 5.05e11T + 1.61e23T^{2} \)
67 \( 1 - 1.07e9T + 5.48e23T^{2} \)
71 \( 1 - 8.42e11T + 1.16e24T^{2} \)
73 \( 1 + 6.23e11T + 1.67e24T^{2} \)
79 \( 1 + 2.92e12T + 4.66e24T^{2} \)
83 \( 1 - 1.48e12T + 8.87e24T^{2} \)
89 \( 1 - 8.15e12T + 2.19e25T^{2} \)
97 \( 1 - 1.16e13T + 6.73e25T^{2} \)
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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

−9.943320121671181102671083303687, −9.178256791291234715594533535212, −7.999537004581089895773223400867, −6.72986479357848325851370214797, −5.99809461973883767582364560584, −4.91701477472603781444612237871, −3.34232052886645526480990380765, −1.97010434659040896607928239136, −0.971203470911935335057166870970, 0, 0.971203470911935335057166870970, 1.97010434659040896607928239136, 3.34232052886645526480990380765, 4.91701477472603781444612237871, 5.99809461973883767582364560584, 6.72986479357848325851370214797, 7.999537004581089895773223400867, 9.178256791291234715594533535212, 9.943320121671181102671083303687

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