Properties

Label 2-177-1.1-c5-0-28
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  − 1.62·2-s − 9·3-s − 29.3·4-s + 103.·5-s + 14.6·6-s − 137.·7-s + 99.6·8-s + 81·9-s − 168.·10-s − 636.·11-s + 264.·12-s + 834.·13-s + 223.·14-s − 931.·15-s + 777.·16-s + 444.·17-s − 131.·18-s − 513.·19-s − 3.03e3·20-s + 1.23e3·21-s + 1.03e3·22-s + 3.56e3·23-s − 896.·24-s + 7.58e3·25-s − 1.35e3·26-s − 729·27-s + 4.03e3·28-s + ⋯
L(s)  = 1  − 0.286·2-s − 0.577·3-s − 0.917·4-s + 1.85·5-s + 0.165·6-s − 1.06·7-s + 0.550·8-s + 0.333·9-s − 0.531·10-s − 1.58·11-s + 0.529·12-s + 1.36·13-s + 0.304·14-s − 1.06·15-s + 0.759·16-s + 0.373·17-s − 0.0956·18-s − 0.326·19-s − 1.69·20-s + 0.612·21-s + 0.455·22-s + 1.40·23-s − 0.317·24-s + 2.42·25-s − 0.393·26-s − 0.192·27-s + 0.973·28-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 \( 1 + 1.62T + 32T^{2} \)
5 \( 1 - 103.T + 3.12e3T^{2} \)
7 \( 1 + 137.T + 1.68e4T^{2} \)
11 \( 1 + 636.T + 1.61e5T^{2} \)
13 \( 1 - 834.T + 3.71e5T^{2} \)
17 \( 1 - 444.T + 1.41e6T^{2} \)
19 \( 1 + 513.T + 2.47e6T^{2} \)
23 \( 1 - 3.56e3T + 6.43e6T^{2} \)
29 \( 1 + 6.87e3T + 2.05e7T^{2} \)
31 \( 1 + 3.63e3T + 2.86e7T^{2} \)
37 \( 1 + 7.91e3T + 6.93e7T^{2} \)
41 \( 1 + 1.99e4T + 1.15e8T^{2} \)
43 \( 1 + 2.56e3T + 1.47e8T^{2} \)
47 \( 1 + 2.22e4T + 2.29e8T^{2} \)
53 \( 1 - 2.99e4T + 4.18e8T^{2} \)
61 \( 1 + 3.45e4T + 8.44e8T^{2} \)
67 \( 1 - 4.93e4T + 1.35e9T^{2} \)
71 \( 1 + 3.24e4T + 1.80e9T^{2} \)
73 \( 1 + 7.83e3T + 2.07e9T^{2} \)
79 \( 1 + 6.92e4T + 3.07e9T^{2} \)
83 \( 1 + 3.04e4T + 3.93e9T^{2} \)
89 \( 1 - 2.01e4T + 5.58e9T^{2} \)
97 \( 1 + 9.16e4T + 8.58e9T^{2} \)
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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

−10.82448371106372691027132684861, −10.21122225114023951663554175452, −9.457210209782953348957533441084, −8.556363064368876338312328639147, −6.88016311660453653505743444450, −5.71187175624424826809195908778, −5.19819714254169550445900259959, −3.25979847504843219389997226897, −1.53298621313985742549708516419, 0, 1.53298621313985742549708516419, 3.25979847504843219389997226897, 5.19819714254169550445900259959, 5.71187175624424826809195908778, 6.88016311660453653505743444450, 8.556363064368876338312328639147, 9.457210209782953348957533441084, 10.21122225114023951663554175452, 10.82448371106372691027132684861

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