Properties

Label 2-177-1.1-c13-0-98
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  + 60.2·2-s + 729·3-s − 4.56e3·4-s − 2.87e3·5-s + 4.39e4·6-s + 1.93e4·7-s − 7.68e5·8-s + 5.31e5·9-s − 1.73e5·10-s + 1.70e6·11-s − 3.32e6·12-s + 4.41e6·13-s + 1.16e6·14-s − 2.09e6·15-s − 8.88e6·16-s + 4.11e7·17-s + 3.20e7·18-s − 1.87e8·19-s + 1.31e7·20-s + 1.40e7·21-s + 1.02e8·22-s + 8.00e8·23-s − 5.60e8·24-s − 1.21e9·25-s + 2.65e8·26-s + 3.87e8·27-s − 8.81e7·28-s + ⋯
L(s)  = 1  + 0.665·2-s + 0.577·3-s − 0.557·4-s − 0.0824·5-s + 0.384·6-s + 0.0620·7-s − 1.03·8-s + 0.333·9-s − 0.0548·10-s + 0.290·11-s − 0.321·12-s + 0.253·13-s + 0.0412·14-s − 0.0475·15-s − 0.132·16-s + 0.413·17-s + 0.221·18-s − 0.912·19-s + 0.0459·20-s + 0.0358·21-s + 0.193·22-s + 1.12·23-s − 0.598·24-s − 0.993·25-s + 0.168·26-s + 0.192·27-s − 0.0345·28-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 \( 1 - 60.2T + 8.19e3T^{2} \)
5 \( 1 + 2.87e3T + 1.22e9T^{2} \)
7 \( 1 - 1.93e4T + 9.68e10T^{2} \)
11 \( 1 - 1.70e6T + 3.45e13T^{2} \)
13 \( 1 - 4.41e6T + 3.02e14T^{2} \)
17 \( 1 - 4.11e7T + 9.90e15T^{2} \)
19 \( 1 + 1.87e8T + 4.20e16T^{2} \)
23 \( 1 - 8.00e8T + 5.04e17T^{2} \)
29 \( 1 - 3.17e9T + 1.02e19T^{2} \)
31 \( 1 + 3.99e9T + 2.44e19T^{2} \)
37 \( 1 + 7.40e9T + 2.43e20T^{2} \)
41 \( 1 - 3.23e10T + 9.25e20T^{2} \)
43 \( 1 + 1.18e9T + 1.71e21T^{2} \)
47 \( 1 - 1.69e10T + 5.46e21T^{2} \)
53 \( 1 + 4.59e10T + 2.60e22T^{2} \)
61 \( 1 - 1.96e11T + 1.61e23T^{2} \)
67 \( 1 + 1.13e12T + 5.48e23T^{2} \)
71 \( 1 - 9.45e11T + 1.16e24T^{2} \)
73 \( 1 + 1.00e12T + 1.67e24T^{2} \)
79 \( 1 - 2.28e12T + 4.66e24T^{2} \)
83 \( 1 + 4.14e12T + 8.87e24T^{2} \)
89 \( 1 - 2.88e12T + 2.19e25T^{2} \)
97 \( 1 + 1.59e13T + 6.73e25T^{2} \)
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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

−9.654346447412160247534004744711, −8.859621430546987088750315263936, −7.990213776872496778492295459370, −6.68437387668168479409799313729, −5.57256478789645809839189276590, −4.47830623511929086979355423361, −3.67210410669868542090380334305, −2.68027601518753127944514337671, −1.29434792181870696870447451618, 0, 1.29434792181870696870447451618, 2.68027601518753127944514337671, 3.67210410669868542090380334305, 4.47830623511929086979355423361, 5.57256478789645809839189276590, 6.68437387668168479409799313729, 7.990213776872496778492295459370, 8.859621430546987088750315263936, 9.654346447412160247534004744711

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