Properties

Label 2-177-1.1-c11-0-60
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 86.0·2-s − 243·3-s + 5.35e3·4-s + 1.28e4·5-s + 2.09e4·6-s − 7.28e4·7-s − 2.84e5·8-s + 5.90e4·9-s − 1.10e6·10-s − 1.28e5·11-s − 1.30e6·12-s + 1.06e6·13-s + 6.26e6·14-s − 3.12e6·15-s + 1.34e7·16-s − 4.24e6·17-s − 5.07e6·18-s + 4.17e6·19-s + 6.88e7·20-s + 1.76e7·21-s + 1.10e7·22-s − 1.73e7·23-s + 6.90e7·24-s + 1.16e8·25-s − 9.17e7·26-s − 1.43e7·27-s − 3.89e8·28-s + ⋯
L(s)  = 1  − 1.90·2-s − 0.577·3-s + 2.61·4-s + 1.84·5-s + 1.09·6-s − 1.63·7-s − 3.06·8-s + 0.333·9-s − 3.50·10-s − 0.240·11-s − 1.50·12-s + 0.796·13-s + 3.11·14-s − 1.06·15-s + 3.21·16-s − 0.725·17-s − 0.633·18-s + 0.386·19-s + 4.81·20-s + 0.945·21-s + 0.456·22-s − 0.563·23-s + 1.76·24-s + 2.39·25-s − 1.51·26-s − 0.192·27-s − 4.27·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 243T \)
59 \( 1 - 7.14e8T \)
good2 \( 1 + 86.0T + 2.04e3T^{2} \)
5 \( 1 - 1.28e4T + 4.88e7T^{2} \)
7 \( 1 + 7.28e4T + 1.97e9T^{2} \)
11 \( 1 + 1.28e5T + 2.85e11T^{2} \)
13 \( 1 - 1.06e6T + 1.79e12T^{2} \)
17 \( 1 + 4.24e6T + 3.42e13T^{2} \)
19 \( 1 - 4.17e6T + 1.16e14T^{2} \)
23 \( 1 + 1.73e7T + 9.52e14T^{2} \)
29 \( 1 - 9.35e7T + 1.22e16T^{2} \)
31 \( 1 - 5.43e7T + 2.54e16T^{2} \)
37 \( 1 + 5.85e7T + 1.77e17T^{2} \)
41 \( 1 + 1.22e9T + 5.50e17T^{2} \)
43 \( 1 + 1.70e9T + 9.29e17T^{2} \)
47 \( 1 - 2.74e9T + 2.47e18T^{2} \)
53 \( 1 - 8.82e8T + 9.26e18T^{2} \)
61 \( 1 + 2.92e9T + 4.35e19T^{2} \)
67 \( 1 - 1.69e10T + 1.22e20T^{2} \)
71 \( 1 + 3.58e9T + 2.31e20T^{2} \)
73 \( 1 + 1.17e10T + 3.13e20T^{2} \)
79 \( 1 + 4.15e10T + 7.47e20T^{2} \)
83 \( 1 + 3.90e10T + 1.28e21T^{2} \)
89 \( 1 - 8.36e10T + 2.77e21T^{2} \)
97 \( 1 - 2.64e10T + 7.15e21T^{2} \)
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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

−10.20544046715964797752034321756, −9.346591052351181551592365978587, −8.590566811795324872446617539781, −6.85192283761378286428907022610, −6.43418319731946982600229177362, −5.64046457588387408815218692084, −3.05350464268018338051034512239, −2.05325582951621343858934294632, −1.05222349898541232778074964791, 0, 1.05222349898541232778074964791, 2.05325582951621343858934294632, 3.05350464268018338051034512239, 5.64046457588387408815218692084, 6.43418319731946982600229177362, 6.85192283761378286428907022610, 8.590566811795324872446617539781, 9.346591052351181551592365978587, 10.20544046715964797752034321756

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