Properties

Label 2-177-1.1-c9-0-14
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  − 43.0·2-s − 81·3-s + 1.34e3·4-s + 1.40e3·5-s + 3.48e3·6-s + 9.39e3·7-s − 3.56e4·8-s + 6.56e3·9-s − 6.05e4·10-s − 9.13e4·11-s − 1.08e5·12-s − 9.59e4·13-s − 4.04e5·14-s − 1.13e5·15-s + 8.49e5·16-s − 2.11e5·17-s − 2.82e5·18-s + 5.41e5·19-s + 1.88e6·20-s − 7.60e5·21-s + 3.93e6·22-s − 1.61e6·23-s + 2.89e6·24-s + 2.65e4·25-s + 4.13e6·26-s − 5.31e5·27-s + 1.25e7·28-s + ⋯
L(s)  = 1  − 1.90·2-s − 0.577·3-s + 2.61·4-s + 1.00·5-s + 1.09·6-s + 1.47·7-s − 3.08·8-s + 0.333·9-s − 1.91·10-s − 1.88·11-s − 1.51·12-s − 0.932·13-s − 2.81·14-s − 0.581·15-s + 3.24·16-s − 0.614·17-s − 0.634·18-s + 0.952·19-s + 2.63·20-s − 0.853·21-s + 3.57·22-s − 1.20·23-s + 1.77·24-s + 0.0135·25-s + 1.77·26-s − 0.192·27-s + 3.87·28-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

\(L(5)\) \(\approx\) \(0.6666952683\)
\(L(\frac12)\) \(\approx\) \(0.6666952683\)
\(L(\frac{11}{2})\) 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 \( 1 + 43.0T + 512T^{2} \)
5 \( 1 - 1.40e3T + 1.95e6T^{2} \)
7 \( 1 - 9.39e3T + 4.03e7T^{2} \)
11 \( 1 + 9.13e4T + 2.35e9T^{2} \)
13 \( 1 + 9.59e4T + 1.06e10T^{2} \)
17 \( 1 + 2.11e5T + 1.18e11T^{2} \)
19 \( 1 - 5.41e5T + 3.22e11T^{2} \)
23 \( 1 + 1.61e6T + 1.80e12T^{2} \)
29 \( 1 + 7.91e5T + 1.45e13T^{2} \)
31 \( 1 + 7.46e6T + 2.64e13T^{2} \)
37 \( 1 - 8.37e6T + 1.29e14T^{2} \)
41 \( 1 - 1.08e7T + 3.27e14T^{2} \)
43 \( 1 + 4.24e6T + 5.02e14T^{2} \)
47 \( 1 - 6.51e7T + 1.11e15T^{2} \)
53 \( 1 + 3.72e7T + 3.29e15T^{2} \)
61 \( 1 + 1.44e8T + 1.16e16T^{2} \)
67 \( 1 - 7.39e7T + 2.72e16T^{2} \)
71 \( 1 + 3.81e7T + 4.58e16T^{2} \)
73 \( 1 - 3.89e8T + 5.88e16T^{2} \)
79 \( 1 - 6.34e8T + 1.19e17T^{2} \)
83 \( 1 + 6.48e8T + 1.86e17T^{2} \)
89 \( 1 - 9.28e8T + 3.50e17T^{2} \)
97 \( 1 + 2.03e8T + 7.60e17T^{2} \)
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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

−10.71989288674378324212732458540, −10.04557401129377681305569880589, −9.149222429996878661416479432928, −7.81893378080837017682121012843, −7.51154298216042148774770211275, −5.92582556129456486780368487542, −5.10521105673947128142684200674, −2.36863404184670826071344498495, −1.86346061290841794659417460557, −0.53497635808471075067805182492, 0.53497635808471075067805182492, 1.86346061290841794659417460557, 2.36863404184670826071344498495, 5.10521105673947128142684200674, 5.92582556129456486780368487542, 7.51154298216042148774770211275, 7.81893378080837017682121012843, 9.149222429996878661416479432928, 10.04557401129377681305569880589, 10.71989288674378324212732458540

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