Properties

Label 2-177-1.1-c13-0-110
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  + 140.·2-s − 729·3-s + 1.16e4·4-s + 2.18e4·5-s − 1.02e5·6-s − 2.79e5·7-s + 4.86e5·8-s + 5.31e5·9-s + 3.08e6·10-s + 1.20e6·11-s − 8.48e6·12-s + 7.07e6·13-s − 3.93e7·14-s − 1.59e7·15-s − 2.69e7·16-s − 3.53e7·17-s + 7.48e7·18-s + 1.67e8·19-s + 2.54e8·20-s + 2.03e8·21-s + 1.70e8·22-s + 6.54e8·23-s − 3.54e8·24-s − 7.41e8·25-s + 9.96e8·26-s − 3.87e8·27-s − 3.25e9·28-s + ⋯
L(s)  = 1  + 1.55·2-s − 0.577·3-s + 1.42·4-s + 0.626·5-s − 0.898·6-s − 0.898·7-s + 0.655·8-s + 0.333·9-s + 0.974·10-s + 0.205·11-s − 0.820·12-s + 0.406·13-s − 1.39·14-s − 0.361·15-s − 0.401·16-s − 0.354·17-s + 0.518·18-s + 0.818·19-s + 0.890·20-s + 0.518·21-s + 0.320·22-s + 0.922·23-s − 0.378·24-s − 0.607·25-s + 0.632·26-s − 0.192·27-s − 1.27·28-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 \( 1 - 140.T + 8.19e3T^{2} \)
5 \( 1 - 2.18e4T + 1.22e9T^{2} \)
7 \( 1 + 2.79e5T + 9.68e10T^{2} \)
11 \( 1 - 1.20e6T + 3.45e13T^{2} \)
13 \( 1 - 7.07e6T + 3.02e14T^{2} \)
17 \( 1 + 3.53e7T + 9.90e15T^{2} \)
19 \( 1 - 1.67e8T + 4.20e16T^{2} \)
23 \( 1 - 6.54e8T + 5.04e17T^{2} \)
29 \( 1 + 4.99e9T + 1.02e19T^{2} \)
31 \( 1 - 7.48e9T + 2.44e19T^{2} \)
37 \( 1 + 1.13e10T + 2.43e20T^{2} \)
41 \( 1 - 3.92e10T + 9.25e20T^{2} \)
43 \( 1 + 4.27e10T + 1.71e21T^{2} \)
47 \( 1 + 3.05e10T + 5.46e21T^{2} \)
53 \( 1 - 3.73e10T + 2.60e22T^{2} \)
61 \( 1 + 3.93e11T + 1.61e23T^{2} \)
67 \( 1 + 1.78e10T + 5.48e23T^{2} \)
71 \( 1 + 1.02e12T + 1.16e24T^{2} \)
73 \( 1 + 1.52e12T + 1.67e24T^{2} \)
79 \( 1 + 5.44e11T + 4.66e24T^{2} \)
83 \( 1 + 5.17e12T + 8.87e24T^{2} \)
89 \( 1 + 1.83e12T + 2.19e25T^{2} \)
97 \( 1 - 1.43e13T + 6.73e25T^{2} \)
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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

−10.01706261396499557964066771893, −9.076624923939504244010016431547, −7.24507988126585858738435945470, −6.30512626768026491160172599681, −5.74241845132513369388072847647, −4.75571458296440232434282128864, −3.66584136185298849569793404046, −2.76988702075221902606317746807, −1.47491158053810276347915453112, 0, 1.47491158053810276347915453112, 2.76988702075221902606317746807, 3.66584136185298849569793404046, 4.75571458296440232434282128864, 5.74241845132513369388072847647, 6.30512626768026491160172599681, 7.24507988126585858738435945470, 9.076624923939504244010016431547, 10.01706261396499557964066771893

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