Properties

Label 2-177-1.1-c7-0-33
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $55.2921$
Root an. cond. $7.43586$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.05·2-s − 27·3-s − 126.·4-s + 151.·5-s + 28.5·6-s − 1.57e3·7-s + 269.·8-s + 729·9-s − 160.·10-s + 1.59e3·11-s + 3.42e3·12-s + 8.75e3·13-s + 1.66e3·14-s − 4.09e3·15-s + 1.59e4·16-s + 1.98e4·17-s − 771.·18-s − 4.83e3·19-s − 1.92e4·20-s + 4.25e4·21-s − 1.69e3·22-s − 3.07e4·23-s − 7.28e3·24-s − 5.51e4·25-s − 9.27e3·26-s − 1.96e4·27-s + 1.99e5·28-s + ⋯
L(s)  = 1  − 0.0935·2-s − 0.577·3-s − 0.991·4-s + 0.542·5-s + 0.0540·6-s − 1.73·7-s + 0.186·8-s + 0.333·9-s − 0.0507·10-s + 0.361·11-s + 0.572·12-s + 1.10·13-s + 0.162·14-s − 0.313·15-s + 0.973·16-s + 0.979·17-s − 0.0311·18-s − 0.161·19-s − 0.537·20-s + 1.00·21-s − 0.0338·22-s − 0.526·23-s − 0.107·24-s − 0.705·25-s − 0.103·26-s − 0.192·27-s + 1.71·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(55.2921\)
Root analytic conductor: \(7.43586\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 27T \)
59 \( 1 - 2.05e5T \)
good2 \( 1 + 1.05T + 128T^{2} \)
5 \( 1 - 151.T + 7.81e4T^{2} \)
7 \( 1 + 1.57e3T + 8.23e5T^{2} \)
11 \( 1 - 1.59e3T + 1.94e7T^{2} \)
13 \( 1 - 8.75e3T + 6.27e7T^{2} \)
17 \( 1 - 1.98e4T + 4.10e8T^{2} \)
19 \( 1 + 4.83e3T + 8.93e8T^{2} \)
23 \( 1 + 3.07e4T + 3.40e9T^{2} \)
29 \( 1 - 1.47e4T + 1.72e10T^{2} \)
31 \( 1 - 7.22e4T + 2.75e10T^{2} \)
37 \( 1 - 3.57e5T + 9.49e10T^{2} \)
41 \( 1 - 5.62e5T + 1.94e11T^{2} \)
43 \( 1 - 6.82e5T + 2.71e11T^{2} \)
47 \( 1 + 1.24e6T + 5.06e11T^{2} \)
53 \( 1 + 1.94e6T + 1.17e12T^{2} \)
61 \( 1 + 2.48e6T + 3.14e12T^{2} \)
67 \( 1 + 1.19e6T + 6.06e12T^{2} \)
71 \( 1 + 5.50e6T + 9.09e12T^{2} \)
73 \( 1 + 1.03e6T + 1.10e13T^{2} \)
79 \( 1 + 1.05e6T + 1.92e13T^{2} \)
83 \( 1 - 8.29e6T + 2.71e13T^{2} \)
89 \( 1 - 1.51e6T + 4.42e13T^{2} \)
97 \( 1 - 1.02e7T + 8.07e13T^{2} \)
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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

−10.64698961468153745034459837615, −9.707937752070995931755915802479, −9.274858057056511396112751097468, −7.85335739167202863945451937503, −6.24361883611850325168382168468, −5.87194526877788367521950236906, −4.25755463369430888850066383461, −3.21431499932930812189016655562, −1.15737645649049497958418711907, 0, 1.15737645649049497958418711907, 3.21431499932930812189016655562, 4.25755463369430888850066383461, 5.87194526877788367521950236906, 6.24361883611850325168382168468, 7.85335739167202863945451937503, 9.274858057056511396112751097468, 9.707937752070995931755915802479, 10.64698961468153745034459837615

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