Properties

Label 2-177-1.1-c13-0-53
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  − 80.4·2-s − 729·3-s − 1.71e3·4-s − 1.90e4·5-s + 5.86e4·6-s − 2.13e5·7-s + 7.97e5·8-s + 5.31e5·9-s + 1.53e6·10-s + 8.17e6·11-s + 1.24e6·12-s + 1.28e7·13-s + 1.71e7·14-s + 1.39e7·15-s − 5.01e7·16-s − 1.73e8·17-s − 4.27e7·18-s − 1.17e8·19-s + 3.26e7·20-s + 1.55e8·21-s − 6.58e8·22-s − 4.16e8·23-s − 5.81e8·24-s − 8.56e8·25-s − 1.03e9·26-s − 3.87e8·27-s + 3.65e8·28-s + ⋯
L(s)  = 1  − 0.889·2-s − 0.577·3-s − 0.208·4-s − 0.545·5-s + 0.513·6-s − 0.686·7-s + 1.07·8-s + 0.333·9-s + 0.485·10-s + 1.39·11-s + 0.120·12-s + 0.737·13-s + 0.610·14-s + 0.315·15-s − 0.747·16-s − 1.74·17-s − 0.296·18-s − 0.575·19-s + 0.114·20-s + 0.396·21-s − 1.23·22-s − 0.586·23-s − 0.620·24-s − 0.701·25-s − 0.655·26-s − 0.192·27-s + 0.143·28-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 \( 1 + 80.4T + 8.19e3T^{2} \)
5 \( 1 + 1.90e4T + 1.22e9T^{2} \)
7 \( 1 + 2.13e5T + 9.68e10T^{2} \)
11 \( 1 - 8.17e6T + 3.45e13T^{2} \)
13 \( 1 - 1.28e7T + 3.02e14T^{2} \)
17 \( 1 + 1.73e8T + 9.90e15T^{2} \)
19 \( 1 + 1.17e8T + 4.20e16T^{2} \)
23 \( 1 + 4.16e8T + 5.04e17T^{2} \)
29 \( 1 + 1.55e8T + 1.02e19T^{2} \)
31 \( 1 + 3.96e9T + 2.44e19T^{2} \)
37 \( 1 - 1.91e10T + 2.43e20T^{2} \)
41 \( 1 - 4.39e10T + 9.25e20T^{2} \)
43 \( 1 - 5.38e10T + 1.71e21T^{2} \)
47 \( 1 + 9.05e10T + 5.46e21T^{2} \)
53 \( 1 - 3.89e10T + 2.60e22T^{2} \)
61 \( 1 - 1.44e11T + 1.61e23T^{2} \)
67 \( 1 - 3.53e11T + 5.48e23T^{2} \)
71 \( 1 - 8.31e11T + 1.16e24T^{2} \)
73 \( 1 - 1.24e11T + 1.67e24T^{2} \)
79 \( 1 - 1.36e12T + 4.66e24T^{2} \)
83 \( 1 + 2.42e12T + 8.87e24T^{2} \)
89 \( 1 - 1.71e12T + 2.19e25T^{2} \)
97 \( 1 - 1.14e13T + 6.73e25T^{2} \)
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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

−9.598588532414572697380317754363, −9.039917601404044100344460956997, −8.003265059902291248013715605010, −6.82803886173078646719316468303, −6.06805212201412867198265967594, −4.36434699744165611721774488043, −3.87573150448617153272810517566, −1.97104879364643632688870604431, −0.814435619741079473914245416501, 0, 0.814435619741079473914245416501, 1.97104879364643632688870604431, 3.87573150448617153272810517566, 4.36434699744165611721774488043, 6.06805212201412867198265967594, 6.82803886173078646719316468303, 8.003265059902291248013715605010, 9.039917601404044100344460956997, 9.598588532414572697380317754363

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