Properties

Label 2-177-1.1-c5-0-47
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.42·2-s + 9·3-s + 38.9·4-s − 30.3·5-s + 75.8·6-s − 197.·7-s + 58.7·8-s + 81·9-s − 255.·10-s − 604.·11-s + 350.·12-s − 893.·13-s − 1.66e3·14-s − 273.·15-s − 752.·16-s + 1.08e3·17-s + 682.·18-s + 2.02e3·19-s − 1.18e3·20-s − 1.77e3·21-s − 5.09e3·22-s + 1.32e3·23-s + 528.·24-s − 2.20e3·25-s − 7.52e3·26-s + 729·27-s − 7.70e3·28-s + ⋯
L(s)  = 1  + 1.48·2-s + 0.577·3-s + 1.21·4-s − 0.543·5-s + 0.859·6-s − 1.52·7-s + 0.324·8-s + 0.333·9-s − 0.809·10-s − 1.50·11-s + 0.703·12-s − 1.46·13-s − 2.27·14-s − 0.313·15-s − 0.734·16-s + 0.913·17-s + 0.496·18-s + 1.28·19-s − 0.661·20-s − 0.880·21-s − 2.24·22-s + 0.521·23-s + 0.187·24-s − 0.704·25-s − 2.18·26-s + 0.192·27-s − 1.85·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 - 8.42T + 32T^{2} \)
5 \( 1 + 30.3T + 3.12e3T^{2} \)
7 \( 1 + 197.T + 1.68e4T^{2} \)
11 \( 1 + 604.T + 1.61e5T^{2} \)
13 \( 1 + 893.T + 3.71e5T^{2} \)
17 \( 1 - 1.08e3T + 1.41e6T^{2} \)
19 \( 1 - 2.02e3T + 2.47e6T^{2} \)
23 \( 1 - 1.32e3T + 6.43e6T^{2} \)
29 \( 1 - 2.51e3T + 2.05e7T^{2} \)
31 \( 1 - 8.78e3T + 2.86e7T^{2} \)
37 \( 1 + 1.30e4T + 6.93e7T^{2} \)
41 \( 1 - 1.06e4T + 1.15e8T^{2} \)
43 \( 1 - 7.39e3T + 1.47e8T^{2} \)
47 \( 1 + 2.34e4T + 2.29e8T^{2} \)
53 \( 1 + 1.74e4T + 4.18e8T^{2} \)
61 \( 1 + 1.17e4T + 8.44e8T^{2} \)
67 \( 1 + 4.69e4T + 1.35e9T^{2} \)
71 \( 1 + 3.46e4T + 1.80e9T^{2} \)
73 \( 1 + 3.11e4T + 2.07e9T^{2} \)
79 \( 1 + 7.21e4T + 3.07e9T^{2} \)
83 \( 1 - 3.14e4T + 3.93e9T^{2} \)
89 \( 1 - 1.01e5T + 5.58e9T^{2} \)
97 \( 1 + 1.82e5T + 8.58e9T^{2} \)
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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

−11.92132643091412496622430194535, −10.25946091780862039779177289654, −9.509307535355125826199026492451, −7.84655299565924140550587119813, −6.98820985084515780331575077900, −5.64762525665602448963489283050, −4.63806443404185257862366316040, −3.17362033918638656828068589940, −2.82636828125798405848457810768, 0, 2.82636828125798405848457810768, 3.17362033918638656828068589940, 4.63806443404185257862366316040, 5.64762525665602448963489283050, 6.98820985084515780331575077900, 7.84655299565924140550587119813, 9.509307535355125826199026492451, 10.25946091780862039779177289654, 11.92132643091412496622430194535

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