Properties

Label 2-177-1.1-c5-0-16
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  − 4.14·2-s − 9·3-s − 14.8·4-s − 34.9·5-s + 37.3·6-s − 110.·7-s + 194.·8-s + 81·9-s + 144.·10-s + 176.·11-s + 133.·12-s + 818.·13-s + 456.·14-s + 314.·15-s − 330.·16-s − 1.40e3·17-s − 335.·18-s + 2.00e3·19-s + 518.·20-s + 992.·21-s − 731.·22-s + 846.·23-s − 1.74e3·24-s − 1.90e3·25-s − 3.39e3·26-s − 729·27-s + 1.63e3·28-s + ⋯
L(s)  = 1  − 0.732·2-s − 0.577·3-s − 0.463·4-s − 0.625·5-s + 0.423·6-s − 0.850·7-s + 1.07·8-s + 0.333·9-s + 0.458·10-s + 0.439·11-s + 0.267·12-s + 1.34·13-s + 0.623·14-s + 0.361·15-s − 0.322·16-s − 1.17·17-s − 0.244·18-s + 1.27·19-s + 0.289·20-s + 0.490·21-s − 0.322·22-s + 0.333·23-s − 0.618·24-s − 0.608·25-s − 0.984·26-s − 0.192·27-s + 0.393·28-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 \( 1 + 4.14T + 32T^{2} \)
5 \( 1 + 34.9T + 3.12e3T^{2} \)
7 \( 1 + 110.T + 1.68e4T^{2} \)
11 \( 1 - 176.T + 1.61e5T^{2} \)
13 \( 1 - 818.T + 3.71e5T^{2} \)
17 \( 1 + 1.40e3T + 1.41e6T^{2} \)
19 \( 1 - 2.00e3T + 2.47e6T^{2} \)
23 \( 1 - 846.T + 6.43e6T^{2} \)
29 \( 1 - 3.64e3T + 2.05e7T^{2} \)
31 \( 1 - 5.74e3T + 2.86e7T^{2} \)
37 \( 1 + 1.23e4T + 6.93e7T^{2} \)
41 \( 1 - 1.19e4T + 1.15e8T^{2} \)
43 \( 1 + 1.37e4T + 1.47e8T^{2} \)
47 \( 1 - 7.63e3T + 2.29e8T^{2} \)
53 \( 1 - 6.03e3T + 4.18e8T^{2} \)
61 \( 1 + 4.02e4T + 8.44e8T^{2} \)
67 \( 1 + 5.42e4T + 1.35e9T^{2} \)
71 \( 1 + 1.52e4T + 1.80e9T^{2} \)
73 \( 1 - 6.86e4T + 2.07e9T^{2} \)
79 \( 1 + 8.50e4T + 3.07e9T^{2} \)
83 \( 1 - 6.23e4T + 3.93e9T^{2} \)
89 \( 1 + 1.36e5T + 5.58e9T^{2} \)
97 \( 1 - 7.75e4T + 8.58e9T^{2} \)
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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

−11.18571750431717269063635706572, −10.23834045895186853798515153230, −9.243790533994873278716774799161, −8.410452142799547128868213382491, −7.18671892250258769450531154681, −6.13207414298376202375727478417, −4.62079440547774134004391208679, −3.51281369855920537011636280744, −1.17231383859425890714673088570, 0, 1.17231383859425890714673088570, 3.51281369855920537011636280744, 4.62079440547774134004391208679, 6.13207414298376202375727478417, 7.18671892250258769450531154681, 8.410452142799547128868213382491, 9.243790533994873278716774799161, 10.23834045895186853798515153230, 11.18571750431717269063635706572

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