Properties

Label 2-177-1.1-c5-0-19
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  + 5.21·2-s + 9·3-s − 4.77·4-s − 21.6·5-s + 46.9·6-s + 150.·7-s − 191.·8-s + 81·9-s − 113.·10-s + 527.·11-s − 42.9·12-s + 51.3·13-s + 783.·14-s − 194.·15-s − 848.·16-s − 968.·17-s + 422.·18-s + 2.27e3·19-s + 103.·20-s + 1.35e3·21-s + 2.75e3·22-s + 3.80e3·23-s − 1.72e3·24-s − 2.65e3·25-s + 268.·26-s + 729·27-s − 716.·28-s + ⋯
L(s)  = 1  + 0.922·2-s + 0.577·3-s − 0.149·4-s − 0.387·5-s + 0.532·6-s + 1.15·7-s − 1.05·8-s + 0.333·9-s − 0.357·10-s + 1.31·11-s − 0.0861·12-s + 0.0843·13-s + 1.06·14-s − 0.223·15-s − 0.828·16-s − 0.812·17-s + 0.307·18-s + 1.44·19-s + 0.0577·20-s + 0.668·21-s + 1.21·22-s + 1.49·23-s − 0.611·24-s − 0.849·25-s + 0.0777·26-s + 0.192·27-s − 0.172·28-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: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.855464633\)
\(L(\frac12)\) \(\approx\) \(3.855464633\)
\(L(\frac{7}{2})\) 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 \( 1 - 5.21T + 32T^{2} \)
5 \( 1 + 21.6T + 3.12e3T^{2} \)
7 \( 1 - 150.T + 1.68e4T^{2} \)
11 \( 1 - 527.T + 1.61e5T^{2} \)
13 \( 1 - 51.3T + 3.71e5T^{2} \)
17 \( 1 + 968.T + 1.41e6T^{2} \)
19 \( 1 - 2.27e3T + 2.47e6T^{2} \)
23 \( 1 - 3.80e3T + 6.43e6T^{2} \)
29 \( 1 - 7.26e3T + 2.05e7T^{2} \)
31 \( 1 + 513.T + 2.86e7T^{2} \)
37 \( 1 - 1.13e4T + 6.93e7T^{2} \)
41 \( 1 - 1.70e4T + 1.15e8T^{2} \)
43 \( 1 + 7.35e3T + 1.47e8T^{2} \)
47 \( 1 + 1.53e4T + 2.29e8T^{2} \)
53 \( 1 + 3.20e4T + 4.18e8T^{2} \)
61 \( 1 - 1.38e4T + 8.44e8T^{2} \)
67 \( 1 + 2.62e4T + 1.35e9T^{2} \)
71 \( 1 + 4.92e4T + 1.80e9T^{2} \)
73 \( 1 + 5.31e3T + 2.07e9T^{2} \)
79 \( 1 + 4.11e4T + 3.07e9T^{2} \)
83 \( 1 - 6.09e4T + 3.93e9T^{2} \)
89 \( 1 - 4.24e3T + 5.58e9T^{2} \)
97 \( 1 + 4.91e3T + 8.58e9T^{2} \)
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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

−11.79770329307389497252593652874, −11.29032003411233671622269902785, −9.538855733110085790133829023523, −8.770187004605072365878792716049, −7.71025252344735671382951893046, −6.39818648496672286499944513796, −4.92369195431987202584210881878, −4.18824586800093705256892171988, −2.98175022519739424329568927292, −1.19122934070908360872753927334, 1.19122934070908360872753927334, 2.98175022519739424329568927292, 4.18824586800093705256892171988, 4.92369195431987202584210881878, 6.39818648496672286499944513796, 7.71025252344735671382951893046, 8.770187004605072365878792716049, 9.538855733110085790133829023523, 11.29032003411233671622269902785, 11.79770329307389497252593652874

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