Properties

Label 2-177-1.1-c13-0-111
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  + 168.·2-s − 729·3-s + 2.02e4·4-s − 5.06e4·5-s − 1.22e5·6-s + 2.03e5·7-s + 2.02e6·8-s + 5.31e5·9-s − 8.54e6·10-s − 3.23e6·11-s − 1.47e7·12-s + 1.43e7·13-s + 3.42e7·14-s + 3.69e7·15-s + 1.75e8·16-s − 9.48e7·17-s + 8.95e7·18-s − 7.38e7·19-s − 1.02e9·20-s − 1.48e8·21-s − 5.44e8·22-s + 2.17e8·23-s − 1.47e9·24-s + 1.34e9·25-s + 2.41e9·26-s − 3.87e8·27-s + 4.10e9·28-s + ⋯
L(s)  = 1  + 1.86·2-s − 0.577·3-s + 2.46·4-s − 1.45·5-s − 1.07·6-s + 0.652·7-s + 2.72·8-s + 0.333·9-s − 2.70·10-s − 0.550·11-s − 1.42·12-s + 0.824·13-s + 1.21·14-s + 0.837·15-s + 2.61·16-s − 0.953·17-s + 0.620·18-s − 0.359·19-s − 3.57·20-s − 0.376·21-s − 1.02·22-s + 0.306·23-s − 1.57·24-s + 1.10·25-s + 1.53·26-s − 0.192·27-s + 1.60·28-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 \( 1 - 168.T + 8.19e3T^{2} \)
5 \( 1 + 5.06e4T + 1.22e9T^{2} \)
7 \( 1 - 2.03e5T + 9.68e10T^{2} \)
11 \( 1 + 3.23e6T + 3.45e13T^{2} \)
13 \( 1 - 1.43e7T + 3.02e14T^{2} \)
17 \( 1 + 9.48e7T + 9.90e15T^{2} \)
19 \( 1 + 7.38e7T + 4.20e16T^{2} \)
23 \( 1 - 2.17e8T + 5.04e17T^{2} \)
29 \( 1 - 1.43e9T + 1.02e19T^{2} \)
31 \( 1 + 2.08e9T + 2.44e19T^{2} \)
37 \( 1 + 8.62e9T + 2.43e20T^{2} \)
41 \( 1 - 1.89e10T + 9.25e20T^{2} \)
43 \( 1 + 2.90e10T + 1.71e21T^{2} \)
47 \( 1 - 3.92e10T + 5.46e21T^{2} \)
53 \( 1 - 1.89e10T + 2.60e22T^{2} \)
61 \( 1 + 3.32e11T + 1.61e23T^{2} \)
67 \( 1 - 4.65e11T + 5.48e23T^{2} \)
71 \( 1 + 1.25e12T + 1.16e24T^{2} \)
73 \( 1 + 4.92e11T + 1.67e24T^{2} \)
79 \( 1 + 1.86e12T + 4.66e24T^{2} \)
83 \( 1 + 5.31e11T + 8.87e24T^{2} \)
89 \( 1 + 9.16e12T + 2.19e25T^{2} \)
97 \( 1 + 5.98e12T + 6.73e25T^{2} \)
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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

−10.71382023058523055875900714901, −8.431597086709317073244650408207, −7.42726932439706376079995003320, −6.55880534281965863574518902468, −5.43515003681295096196461686999, −4.51868582657865484934049213326, −3.95415967349468284081938747393, −2.84390108913191062328585931169, −1.49740868921118380096545498133, 0, 1.49740868921118380096545498133, 2.84390108913191062328585931169, 3.95415967349468284081938747393, 4.51868582657865484934049213326, 5.43515003681295096196461686999, 6.55880534281965863574518902468, 7.42726932439706376079995003320, 8.431597086709317073244650408207, 10.71382023058523055875900714901

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