Properties

Label 2-177-1.1-c9-0-43
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 23.3·2-s − 81·3-s + 33.7·4-s + 477.·5-s + 1.89e3·6-s + 1.57e3·7-s + 1.11e4·8-s + 6.56e3·9-s − 1.11e4·10-s − 4.81e4·11-s − 2.73e3·12-s + 6.00e4·13-s − 3.66e4·14-s − 3.86e4·15-s − 2.78e5·16-s + 6.59e4·17-s − 1.53e5·18-s − 3.14e5·19-s + 1.61e4·20-s − 1.27e5·21-s + 1.12e6·22-s − 1.21e6·23-s − 9.04e5·24-s − 1.72e6·25-s − 1.40e6·26-s − 5.31e5·27-s + 5.29e4·28-s + ⋯
L(s)  = 1  − 1.03·2-s − 0.577·3-s + 0.0658·4-s + 0.341·5-s + 0.596·6-s + 0.247·7-s + 0.964·8-s + 0.333·9-s − 0.352·10-s − 0.991·11-s − 0.0380·12-s + 0.583·13-s − 0.255·14-s − 0.197·15-s − 1.06·16-s + 0.191·17-s − 0.344·18-s − 0.553·19-s + 0.0225·20-s − 0.142·21-s + 1.02·22-s − 0.903·23-s − 0.556·24-s − 0.883·25-s − 0.601·26-s − 0.192·27-s + 0.0162·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 81T \)
59 \( 1 + 1.21e7T \)
good2 \( 1 + 23.3T + 512T^{2} \)
5 \( 1 - 477.T + 1.95e6T^{2} \)
7 \( 1 - 1.57e3T + 4.03e7T^{2} \)
11 \( 1 + 4.81e4T + 2.35e9T^{2} \)
13 \( 1 - 6.00e4T + 1.06e10T^{2} \)
17 \( 1 - 6.59e4T + 1.18e11T^{2} \)
19 \( 1 + 3.14e5T + 3.22e11T^{2} \)
23 \( 1 + 1.21e6T + 1.80e12T^{2} \)
29 \( 1 - 5.11e6T + 1.45e13T^{2} \)
31 \( 1 - 2.37e6T + 2.64e13T^{2} \)
37 \( 1 - 1.04e6T + 1.29e14T^{2} \)
41 \( 1 + 9.06e6T + 3.27e14T^{2} \)
43 \( 1 - 3.58e7T + 5.02e14T^{2} \)
47 \( 1 - 3.00e7T + 1.11e15T^{2} \)
53 \( 1 + 2.45e7T + 3.29e15T^{2} \)
61 \( 1 - 6.49e7T + 1.16e16T^{2} \)
67 \( 1 - 2.82e8T + 2.72e16T^{2} \)
71 \( 1 + 6.33e7T + 4.58e16T^{2} \)
73 \( 1 - 3.82e8T + 5.88e16T^{2} \)
79 \( 1 + 5.56e8T + 1.19e17T^{2} \)
83 \( 1 - 4.81e7T + 1.86e17T^{2} \)
89 \( 1 - 1.06e9T + 3.50e17T^{2} \)
97 \( 1 - 1.41e6T + 7.60e17T^{2} \)
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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

−10.37904424052195451030272224194, −9.663046600505854776330623196773, −8.429366594356897154488220998895, −7.75598451759665934589009482191, −6.41729458584913400921749728144, −5.28575751679667213348619841085, −4.15282637090393797437348338379, −2.26451900271015122258432521435, −1.06081978968230649626726265264, 0, 1.06081978968230649626726265264, 2.26451900271015122258432521435, 4.15282637090393797437348338379, 5.28575751679667213348619841085, 6.41729458584913400921749728144, 7.75598451759665934589009482191, 8.429366594356897154488220998895, 9.663046600505854776330623196773, 10.37904424052195451030272224194

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