Properties

Label 2-177-1.1-c9-0-64
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 39.3·2-s − 81·3-s + 1.03e3·4-s + 1.74e3·5-s − 3.18e3·6-s + 7.50e3·7-s + 2.07e4·8-s + 6.56e3·9-s + 6.86e4·10-s + 2.27e3·11-s − 8.41e4·12-s + 8.93e4·13-s + 2.95e5·14-s − 1.41e5·15-s + 2.85e5·16-s + 5.35e5·17-s + 2.58e5·18-s − 1.00e6·19-s + 1.81e6·20-s − 6.07e5·21-s + 8.94e4·22-s + 1.67e6·23-s − 1.68e6·24-s + 1.08e6·25-s + 3.51e6·26-s − 5.31e5·27-s + 7.79e6·28-s + ⋯
L(s)  = 1  + 1.74·2-s − 0.577·3-s + 2.02·4-s + 1.24·5-s − 1.00·6-s + 1.18·7-s + 1.79·8-s + 0.333·9-s + 2.17·10-s + 0.0467·11-s − 1.17·12-s + 0.867·13-s + 2.05·14-s − 0.720·15-s + 1.08·16-s + 1.55·17-s + 0.580·18-s − 1.77·19-s + 2.53·20-s − 0.681·21-s + 0.0813·22-s + 1.24·23-s − 1.03·24-s + 0.556·25-s + 1.50·26-s − 0.192·27-s + 2.39·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(8.648132438\)
\(L(\frac12)\) \(\approx\) \(8.648132438\)
\(L(\frac{11}{2})\) 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 \( 1 - 39.3T + 512T^{2} \)
5 \( 1 - 1.74e3T + 1.95e6T^{2} \)
7 \( 1 - 7.50e3T + 4.03e7T^{2} \)
11 \( 1 - 2.27e3T + 2.35e9T^{2} \)
13 \( 1 - 8.93e4T + 1.06e10T^{2} \)
17 \( 1 - 5.35e5T + 1.18e11T^{2} \)
19 \( 1 + 1.00e6T + 3.22e11T^{2} \)
23 \( 1 - 1.67e6T + 1.80e12T^{2} \)
29 \( 1 + 1.20e6T + 1.45e13T^{2} \)
31 \( 1 + 8.78e6T + 2.64e13T^{2} \)
37 \( 1 + 1.85e7T + 1.29e14T^{2} \)
41 \( 1 - 3.35e7T + 3.27e14T^{2} \)
43 \( 1 - 2.69e7T + 5.02e14T^{2} \)
47 \( 1 - 1.61e7T + 1.11e15T^{2} \)
53 \( 1 + 3.89e7T + 3.29e15T^{2} \)
61 \( 1 + 1.72e8T + 1.16e16T^{2} \)
67 \( 1 + 8.66e7T + 2.72e16T^{2} \)
71 \( 1 - 2.50e8T + 4.58e16T^{2} \)
73 \( 1 + 4.77e7T + 5.88e16T^{2} \)
79 \( 1 - 5.50e8T + 1.19e17T^{2} \)
83 \( 1 - 6.47e8T + 1.86e17T^{2} \)
89 \( 1 + 7.22e8T + 3.50e17T^{2} \)
97 \( 1 - 5.53e8T + 7.60e17T^{2} \)
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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

−10.98430486438609669162575677273, −10.75752001466425430623444481850, −9.070336436397461811331415816402, −7.48591136027160224629235607239, −6.22713406348288973265078200273, −5.61415931069208026049732252924, −4.84558479469383373270395734506, −3.67473592596106847721146062580, −2.17305024247224151534387906841, −1.34261806809496649004310475679, 1.34261806809496649004310475679, 2.17305024247224151534387906841, 3.67473592596106847721146062580, 4.84558479469383373270395734506, 5.61415931069208026049732252924, 6.22713406348288973265078200273, 7.48591136027160224629235607239, 9.070336436397461811331415816402, 10.75752001466425430623444481850, 10.98430486438609669162575677273

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