Properties

Label 2-177-1.1-c13-0-66
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  − 42.0·2-s − 729·3-s − 6.42e3·4-s − 2.96e4·5-s + 3.06e4·6-s + 5.19e5·7-s + 6.14e5·8-s + 5.31e5·9-s + 1.24e6·10-s + 5.17e6·11-s + 4.68e6·12-s − 1.50e7·13-s − 2.18e7·14-s + 2.16e7·15-s + 2.67e7·16-s − 4.83e7·17-s − 2.23e7·18-s − 1.21e8·19-s + 1.90e8·20-s − 3.78e8·21-s − 2.17e8·22-s + 1.03e6·23-s − 4.48e8·24-s − 3.41e8·25-s + 6.34e8·26-s − 3.87e8·27-s − 3.33e9·28-s + ⋯
L(s)  = 1  − 0.464·2-s − 0.577·3-s − 0.783·4-s − 0.848·5-s + 0.268·6-s + 1.66·7-s + 0.829·8-s + 0.333·9-s + 0.394·10-s + 0.880·11-s + 0.452·12-s − 0.866·13-s − 0.775·14-s + 0.489·15-s + 0.398·16-s − 0.485·17-s − 0.154·18-s − 0.592·19-s + 0.665·20-s − 0.963·21-s − 0.409·22-s + 0.00145·23-s − 0.478·24-s − 0.279·25-s + 0.402·26-s − 0.192·27-s − 1.30·28-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 \( 1 + 42.0T + 8.19e3T^{2} \)
5 \( 1 + 2.96e4T + 1.22e9T^{2} \)
7 \( 1 - 5.19e5T + 9.68e10T^{2} \)
11 \( 1 - 5.17e6T + 3.45e13T^{2} \)
13 \( 1 + 1.50e7T + 3.02e14T^{2} \)
17 \( 1 + 4.83e7T + 9.90e15T^{2} \)
19 \( 1 + 1.21e8T + 4.20e16T^{2} \)
23 \( 1 - 1.03e6T + 5.04e17T^{2} \)
29 \( 1 + 2.75e9T + 1.02e19T^{2} \)
31 \( 1 + 7.52e8T + 2.44e19T^{2} \)
37 \( 1 - 2.97e10T + 2.43e20T^{2} \)
41 \( 1 + 2.00e10T + 9.25e20T^{2} \)
43 \( 1 + 4.09e10T + 1.71e21T^{2} \)
47 \( 1 - 1.04e11T + 5.46e21T^{2} \)
53 \( 1 - 2.07e11T + 2.60e22T^{2} \)
61 \( 1 + 1.41e11T + 1.61e23T^{2} \)
67 \( 1 - 7.07e11T + 5.48e23T^{2} \)
71 \( 1 - 1.32e9T + 1.16e24T^{2} \)
73 \( 1 - 6.69e11T + 1.67e24T^{2} \)
79 \( 1 + 3.83e11T + 4.66e24T^{2} \)
83 \( 1 + 3.96e12T + 8.87e24T^{2} \)
89 \( 1 + 2.74e12T + 2.19e25T^{2} \)
97 \( 1 + 1.54e13T + 6.73e25T^{2} \)
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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

−9.807589719526389305961561828624, −8.723297495566967818901701328840, −7.942977458973856969985828184832, −7.13736554774205028254820522672, −5.51823383080866867020799214864, −4.49853093752911463855698888982, −4.05899260383280713545641660168, −1.97975408375968433225706318835, −0.963674242043585493138541583075, 0, 0.963674242043585493138541583075, 1.97975408375968433225706318835, 4.05899260383280713545641660168, 4.49853093752911463855698888982, 5.51823383080866867020799214864, 7.13736554774205028254820522672, 7.942977458973856969985828184832, 8.723297495566967818901701328840, 9.807589719526389305961561828624

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