Properties

Label 2-177-1.1-c7-0-47
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $55.2921$
Root an. cond. $7.43586$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 18.9·2-s + 27·3-s + 230.·4-s + 260.·5-s + 511.·6-s + 254.·7-s + 1.95e3·8-s + 729·9-s + 4.93e3·10-s − 3.60e3·11-s + 6.23e3·12-s + 1.00e4·13-s + 4.83e3·14-s + 7.03e3·15-s + 7.38e3·16-s + 4.42e3·17-s + 1.38e4·18-s + 1.69e4·19-s + 6.01e4·20-s + 6.88e3·21-s − 6.82e4·22-s + 6.31e4·23-s + 5.26e4·24-s − 1.02e4·25-s + 1.90e5·26-s + 1.96e4·27-s + 5.88e4·28-s + ⋯
L(s)  = 1  + 1.67·2-s + 0.577·3-s + 1.80·4-s + 0.932·5-s + 0.966·6-s + 0.280·7-s + 1.34·8-s + 0.333·9-s + 1.56·10-s − 0.815·11-s + 1.04·12-s + 1.27·13-s + 0.470·14-s + 0.538·15-s + 0.451·16-s + 0.218·17-s + 0.558·18-s + 0.567·19-s + 1.68·20-s + 0.162·21-s − 1.36·22-s + 1.08·23-s + 0.777·24-s − 0.130·25-s + 2.13·26-s + 0.192·27-s + 0.506·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(55.2921\)
Root analytic conductor: \(7.43586\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(8.884541399\)
\(L(\frac12)\) \(\approx\) \(8.884541399\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 27T \)
59 \( 1 - 2.05e5T \)
good2 \( 1 - 18.9T + 128T^{2} \)
5 \( 1 - 260.T + 7.81e4T^{2} \)
7 \( 1 - 254.T + 8.23e5T^{2} \)
11 \( 1 + 3.60e3T + 1.94e7T^{2} \)
13 \( 1 - 1.00e4T + 6.27e7T^{2} \)
17 \( 1 - 4.42e3T + 4.10e8T^{2} \)
19 \( 1 - 1.69e4T + 8.93e8T^{2} \)
23 \( 1 - 6.31e4T + 3.40e9T^{2} \)
29 \( 1 + 2.74e4T + 1.72e10T^{2} \)
31 \( 1 + 2.22e4T + 2.75e10T^{2} \)
37 \( 1 - 1.11e5T + 9.49e10T^{2} \)
41 \( 1 - 8.75e4T + 1.94e11T^{2} \)
43 \( 1 + 4.54e5T + 2.71e11T^{2} \)
47 \( 1 + 7.47e5T + 5.06e11T^{2} \)
53 \( 1 - 1.10e6T + 1.17e12T^{2} \)
61 \( 1 + 2.27e5T + 3.14e12T^{2} \)
67 \( 1 - 1.66e6T + 6.06e12T^{2} \)
71 \( 1 - 1.63e6T + 9.09e12T^{2} \)
73 \( 1 + 3.79e5T + 1.10e13T^{2} \)
79 \( 1 + 1.69e6T + 1.92e13T^{2} \)
83 \( 1 - 3.64e6T + 2.71e13T^{2} \)
89 \( 1 + 7.90e6T + 4.42e13T^{2} \)
97 \( 1 + 9.05e5T + 8.07e13T^{2} \)
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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

−11.59145386359871873621659493351, −10.68886410317130818325136248563, −9.468982028171236275043190761547, −8.173481157558499938719573655388, −6.84687390703526958880249437458, −5.78099632398999030744189976829, −4.98674252709582319900828784963, −3.64848970097063514893637550252, −2.67978756296869028241393512613, −1.51432176893645747567166793556, 1.51432176893645747567166793556, 2.67978756296869028241393512613, 3.64848970097063514893637550252, 4.98674252709582319900828784963, 5.78099632398999030744189976829, 6.84687390703526958880249437458, 8.173481157558499938719573655388, 9.468982028171236275043190761547, 10.68886410317130818325136248563, 11.59145386359871873621659493351

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