Properties

Label 2-177-1.1-c13-0-65
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  + 39.4·2-s − 729·3-s − 6.63e3·4-s + 3.97e4·5-s − 2.87e4·6-s − 5.39e5·7-s − 5.85e5·8-s + 5.31e5·9-s + 1.56e6·10-s − 1.01e7·11-s + 4.83e6·12-s + 1.08e6·13-s − 2.12e7·14-s − 2.89e7·15-s + 3.12e7·16-s + 1.61e8·17-s + 2.09e7·18-s + 1.53e8·19-s − 2.63e8·20-s + 3.92e8·21-s − 4.01e8·22-s + 1.08e8·23-s + 4.26e8·24-s + 3.59e8·25-s + 4.28e7·26-s − 3.87e8·27-s + 3.57e9·28-s + ⋯
L(s)  = 1  + 0.436·2-s − 0.577·3-s − 0.809·4-s + 1.13·5-s − 0.251·6-s − 1.73·7-s − 0.789·8-s + 0.333·9-s + 0.496·10-s − 1.73·11-s + 0.467·12-s + 0.0623·13-s − 0.755·14-s − 0.656·15-s + 0.465·16-s + 1.62·17-s + 0.145·18-s + 0.749·19-s − 0.921·20-s + 0.999·21-s − 0.755·22-s + 0.152·23-s + 0.455·24-s + 0.294·25-s + 0.0272·26-s − 0.192·27-s + 1.40·28-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 \( 1 - 39.4T + 8.19e3T^{2} \)
5 \( 1 - 3.97e4T + 1.22e9T^{2} \)
7 \( 1 + 5.39e5T + 9.68e10T^{2} \)
11 \( 1 + 1.01e7T + 3.45e13T^{2} \)
13 \( 1 - 1.08e6T + 3.02e14T^{2} \)
17 \( 1 - 1.61e8T + 9.90e15T^{2} \)
19 \( 1 - 1.53e8T + 4.20e16T^{2} \)
23 \( 1 - 1.08e8T + 5.04e17T^{2} \)
29 \( 1 - 5.15e9T + 1.02e19T^{2} \)
31 \( 1 + 1.74e8T + 2.44e19T^{2} \)
37 \( 1 + 2.19e10T + 2.43e20T^{2} \)
41 \( 1 - 3.56e10T + 9.25e20T^{2} \)
43 \( 1 - 2.18e10T + 1.71e21T^{2} \)
47 \( 1 + 5.63e10T + 5.46e21T^{2} \)
53 \( 1 + 2.33e10T + 2.60e22T^{2} \)
61 \( 1 + 6.59e11T + 1.61e23T^{2} \)
67 \( 1 - 1.98e10T + 5.48e23T^{2} \)
71 \( 1 - 1.48e12T + 1.16e24T^{2} \)
73 \( 1 + 2.63e11T + 1.67e24T^{2} \)
79 \( 1 - 2.47e12T + 4.66e24T^{2} \)
83 \( 1 + 1.76e12T + 8.87e24T^{2} \)
89 \( 1 - 1.27e12T + 2.19e25T^{2} \)
97 \( 1 - 3.89e12T + 6.73e25T^{2} \)
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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

−10.00329803724735271439866380898, −9.213657347623704187140800146640, −7.75397450350851236408689121694, −6.35787066773923772879328276769, −5.64087550161930099893716870269, −5.02473379644900933610742305170, −3.41820052465532976306619455305, −2.70972477746377144787114822868, −0.942282758074902984526175725109, 0, 0.942282758074902984526175725109, 2.70972477746377144787114822868, 3.41820052465532976306619455305, 5.02473379644900933610742305170, 5.64087550161930099893716870269, 6.35787066773923772879328276769, 7.75397450350851236408689121694, 9.213657347623704187140800146640, 10.00329803724735271439866380898

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