Properties

Label 2-177-1.1-c5-0-10
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.96·2-s − 9·3-s − 16.2·4-s + 45.8·5-s + 35.6·6-s + 214.·7-s + 191.·8-s + 81·9-s − 181.·10-s − 156.·11-s + 146.·12-s + 176.·13-s − 852.·14-s − 412.·15-s − 238.·16-s − 992.·17-s − 321.·18-s − 1.34e3·19-s − 745.·20-s − 1.93e3·21-s + 618.·22-s + 3.04e3·23-s − 1.72e3·24-s − 1.02e3·25-s − 698.·26-s − 729·27-s − 3.49e3·28-s + ⋯
L(s)  = 1  − 0.701·2-s − 0.577·3-s − 0.508·4-s + 0.819·5-s + 0.404·6-s + 1.65·7-s + 1.05·8-s + 0.333·9-s − 0.574·10-s − 0.388·11-s + 0.293·12-s + 0.289·13-s − 1.16·14-s − 0.473·15-s − 0.232·16-s − 0.833·17-s − 0.233·18-s − 0.855·19-s − 0.416·20-s − 0.957·21-s + 0.272·22-s + 1.19·23-s − 0.610·24-s − 0.328·25-s − 0.202·26-s − 0.192·27-s − 0.843·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.315071919\)
\(L(\frac12)\) \(\approx\) \(1.315071919\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9T \)
59 \( 1 - 3.48e3T \)
good2 \( 1 + 3.96T + 32T^{2} \)
5 \( 1 - 45.8T + 3.12e3T^{2} \)
7 \( 1 - 214.T + 1.68e4T^{2} \)
11 \( 1 + 156.T + 1.61e5T^{2} \)
13 \( 1 - 176.T + 3.71e5T^{2} \)
17 \( 1 + 992.T + 1.41e6T^{2} \)
19 \( 1 + 1.34e3T + 2.47e6T^{2} \)
23 \( 1 - 3.04e3T + 6.43e6T^{2} \)
29 \( 1 - 7.85e3T + 2.05e7T^{2} \)
31 \( 1 - 2.34e3T + 2.86e7T^{2} \)
37 \( 1 - 1.09e4T + 6.93e7T^{2} \)
41 \( 1 + 8.94e3T + 1.15e8T^{2} \)
43 \( 1 + 6.62e3T + 1.47e8T^{2} \)
47 \( 1 + 3.59e3T + 2.29e8T^{2} \)
53 \( 1 + 2.99e4T + 4.18e8T^{2} \)
61 \( 1 - 5.09e4T + 8.44e8T^{2} \)
67 \( 1 - 2.01e4T + 1.35e9T^{2} \)
71 \( 1 - 3.72e4T + 1.80e9T^{2} \)
73 \( 1 - 3.24e4T + 2.07e9T^{2} \)
79 \( 1 - 4.72e4T + 3.07e9T^{2} \)
83 \( 1 + 5.52e3T + 3.93e9T^{2} \)
89 \( 1 + 7.63e4T + 5.58e9T^{2} \)
97 \( 1 - 8.09e4T + 8.58e9T^{2} \)
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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

−11.41839893623785463219535274166, −10.74911231995618413448661601415, −9.859653841493852940496321391767, −8.669951655138055625222664847813, −8.007616702105701734329570775373, −6.57678091354293285726401611354, −5.15022457436801557254356318819, −4.51657752069108224336575288818, −2.02102192882851117687702967181, −0.884651381840147443018739709513, 0.884651381840147443018739709513, 2.02102192882851117687702967181, 4.51657752069108224336575288818, 5.15022457436801557254356318819, 6.57678091354293285726401611354, 8.007616702105701734329570775373, 8.669951655138055625222664847813, 9.859653841493852940496321391767, 10.74911231995618413448661601415, 11.41839893623785463219535274166

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