Properties

Label 2-177-1.1-c13-0-118
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  + 149.·2-s + 729·3-s + 1.40e4·4-s − 1.14e4·5-s + 1.08e5·6-s − 1.01e5·7-s + 8.80e5·8-s + 5.31e5·9-s − 1.71e6·10-s − 2.36e6·11-s + 1.02e7·12-s − 6.34e6·13-s − 1.52e7·14-s − 8.37e6·15-s + 1.60e7·16-s − 4.44e7·17-s + 7.93e7·18-s + 9.33e7·19-s − 1.61e8·20-s − 7.43e7·21-s − 3.52e8·22-s + 9.82e8·23-s + 6.42e8·24-s − 1.08e9·25-s − 9.47e8·26-s + 3.87e8·27-s − 1.43e9·28-s + ⋯
L(s)  = 1  + 1.64·2-s + 0.577·3-s + 1.72·4-s − 0.328·5-s + 0.952·6-s − 0.327·7-s + 1.18·8-s + 0.333·9-s − 0.542·10-s − 0.401·11-s + 0.993·12-s − 0.364·13-s − 0.540·14-s − 0.189·15-s + 0.239·16-s − 0.446·17-s + 0.549·18-s + 0.455·19-s − 0.565·20-s − 0.189·21-s − 0.662·22-s + 1.38·23-s + 0.685·24-s − 0.891·25-s − 0.601·26-s + 0.192·27-s − 0.563·28-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 \( 1 - 149.T + 8.19e3T^{2} \)
5 \( 1 + 1.14e4T + 1.22e9T^{2} \)
7 \( 1 + 1.01e5T + 9.68e10T^{2} \)
11 \( 1 + 2.36e6T + 3.45e13T^{2} \)
13 \( 1 + 6.34e6T + 3.02e14T^{2} \)
17 \( 1 + 4.44e7T + 9.90e15T^{2} \)
19 \( 1 - 9.33e7T + 4.20e16T^{2} \)
23 \( 1 - 9.82e8T + 5.04e17T^{2} \)
29 \( 1 + 1.85e8T + 1.02e19T^{2} \)
31 \( 1 + 3.83e9T + 2.44e19T^{2} \)
37 \( 1 - 3.78e9T + 2.43e20T^{2} \)
41 \( 1 + 4.42e10T + 9.25e20T^{2} \)
43 \( 1 + 5.12e10T + 1.71e21T^{2} \)
47 \( 1 + 3.85e10T + 5.46e21T^{2} \)
53 \( 1 - 1.75e11T + 2.60e22T^{2} \)
61 \( 1 + 2.53e11T + 1.61e23T^{2} \)
67 \( 1 - 4.51e11T + 5.48e23T^{2} \)
71 \( 1 + 6.72e11T + 1.16e24T^{2} \)
73 \( 1 + 2.04e12T + 1.67e24T^{2} \)
79 \( 1 + 3.08e12T + 4.66e24T^{2} \)
83 \( 1 - 8.36e11T + 8.87e24T^{2} \)
89 \( 1 - 4.27e12T + 2.19e25T^{2} \)
97 \( 1 + 1.02e13T + 6.73e25T^{2} \)
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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

−9.982712842015269513282105159493, −8.771918362104947302807715652164, −7.47514011891905024437626776085, −6.67588042982255979721460079754, −5.44211432218175742702671402532, −4.59590595591071627367325519197, −3.52813670384035586139218406205, −2.87008042496707256749735367068, −1.75195139194142525843458043188, 0, 1.75195139194142525843458043188, 2.87008042496707256749735367068, 3.52813670384035586139218406205, 4.59590595591071627367325519197, 5.44211432218175742702671402532, 6.67588042982255979721460079754, 7.47514011891905024437626776085, 8.771918362104947302807715652164, 9.982712842015269513282105159493

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