Properties

Label 2-177-1.1-c13-0-85
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  − 159.·2-s − 729·3-s + 1.73e4·4-s − 1.07e4·5-s + 1.16e5·6-s + 1.88e5·7-s − 1.45e6·8-s + 5.31e5·9-s + 1.72e6·10-s + 7.69e6·11-s − 1.26e7·12-s + 8.10e6·13-s − 3.01e7·14-s + 7.86e6·15-s + 9.05e7·16-s − 3.84e7·17-s − 8.48e7·18-s + 2.90e8·19-s − 1.86e8·20-s − 1.37e8·21-s − 1.22e9·22-s + 6.02e8·23-s + 1.06e9·24-s − 1.10e9·25-s − 1.29e9·26-s − 3.87e8·27-s + 3.26e9·28-s + ⋯
L(s)  = 1  − 1.76·2-s − 0.577·3-s + 2.11·4-s − 0.308·5-s + 1.01·6-s + 0.606·7-s − 1.96·8-s + 0.333·9-s + 0.544·10-s + 1.31·11-s − 1.21·12-s + 0.465·13-s − 1.07·14-s + 0.178·15-s + 1.34·16-s − 0.386·17-s − 0.588·18-s + 1.41·19-s − 0.652·20-s − 0.350·21-s − 2.31·22-s + 0.849·23-s + 1.13·24-s − 0.904·25-s − 0.821·26-s − 0.192·27-s + 1.28·28-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 \( 1 + 159.T + 8.19e3T^{2} \)
5 \( 1 + 1.07e4T + 1.22e9T^{2} \)
7 \( 1 - 1.88e5T + 9.68e10T^{2} \)
11 \( 1 - 7.69e6T + 3.45e13T^{2} \)
13 \( 1 - 8.10e6T + 3.02e14T^{2} \)
17 \( 1 + 3.84e7T + 9.90e15T^{2} \)
19 \( 1 - 2.90e8T + 4.20e16T^{2} \)
23 \( 1 - 6.02e8T + 5.04e17T^{2} \)
29 \( 1 - 6.20e9T + 1.02e19T^{2} \)
31 \( 1 + 4.12e9T + 2.44e19T^{2} \)
37 \( 1 - 2.05e9T + 2.43e20T^{2} \)
41 \( 1 + 1.51e10T + 9.25e20T^{2} \)
43 \( 1 + 8.13e10T + 1.71e21T^{2} \)
47 \( 1 + 3.75e10T + 5.46e21T^{2} \)
53 \( 1 + 4.24e10T + 2.60e22T^{2} \)
61 \( 1 + 7.71e11T + 1.61e23T^{2} \)
67 \( 1 - 2.56e10T + 5.48e23T^{2} \)
71 \( 1 + 1.41e12T + 1.16e24T^{2} \)
73 \( 1 + 1.45e12T + 1.67e24T^{2} \)
79 \( 1 - 3.78e12T + 4.66e24T^{2} \)
83 \( 1 + 1.30e12T + 8.87e24T^{2} \)
89 \( 1 - 8.93e11T + 2.19e25T^{2} \)
97 \( 1 - 9.58e12T + 6.73e25T^{2} \)
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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

−9.730189554576916730923591765020, −8.904217831724836807970116167017, −8.009959198018037054162435626519, −7.04367208879114864511486647512, −6.25349725221403567633548008996, −4.76413852143012168999138344832, −3.25835398876787671045076818349, −1.61177423098801115581041981698, −1.12352296850993145045992521653, 0, 1.12352296850993145045992521653, 1.61177423098801115581041981698, 3.25835398876787671045076818349, 4.76413852143012168999138344832, 6.25349725221403567633548008996, 7.04367208879114864511486647512, 8.009959198018037054162435626519, 8.904217831724836807970116167017, 9.730189554576916730923591765020

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