Properties

Label 2-177-1.1-c5-0-22
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  − 6.62·2-s + 9·3-s + 11.8·4-s − 85.2·5-s − 59.6·6-s − 103.·7-s + 133.·8-s + 81·9-s + 564.·10-s + 579.·11-s + 107.·12-s + 435.·13-s + 682.·14-s − 766.·15-s − 1.26e3·16-s + 424.·17-s − 536.·18-s + 1.54e3·19-s − 1.01e3·20-s − 927.·21-s − 3.84e3·22-s − 4.16e3·23-s + 1.19e3·24-s + 4.13e3·25-s − 2.88e3·26-s + 729·27-s − 1.22e3·28-s + ⋯
L(s)  = 1  − 1.17·2-s + 0.577·3-s + 0.371·4-s − 1.52·5-s − 0.676·6-s − 0.794·7-s + 0.735·8-s + 0.333·9-s + 1.78·10-s + 1.44·11-s + 0.214·12-s + 0.714·13-s + 0.930·14-s − 0.879·15-s − 1.23·16-s + 0.355·17-s − 0.390·18-s + 0.979·19-s − 0.566·20-s − 0.458·21-s − 1.69·22-s − 1.64·23-s + 0.424·24-s + 1.32·25-s − 0.837·26-s + 0.192·27-s − 0.295·28-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 \( 1 + 6.62T + 32T^{2} \)
5 \( 1 + 85.2T + 3.12e3T^{2} \)
7 \( 1 + 103.T + 1.68e4T^{2} \)
11 \( 1 - 579.T + 1.61e5T^{2} \)
13 \( 1 - 435.T + 3.71e5T^{2} \)
17 \( 1 - 424.T + 1.41e6T^{2} \)
19 \( 1 - 1.54e3T + 2.47e6T^{2} \)
23 \( 1 + 4.16e3T + 6.43e6T^{2} \)
29 \( 1 + 8.41e3T + 2.05e7T^{2} \)
31 \( 1 - 7.07e3T + 2.86e7T^{2} \)
37 \( 1 - 1.24e4T + 6.93e7T^{2} \)
41 \( 1 + 2.42e3T + 1.15e8T^{2} \)
43 \( 1 + 1.17e4T + 1.47e8T^{2} \)
47 \( 1 + 9.43e3T + 2.29e8T^{2} \)
53 \( 1 + 2.37e4T + 4.18e8T^{2} \)
61 \( 1 - 5.56e3T + 8.44e8T^{2} \)
67 \( 1 + 5.78e4T + 1.35e9T^{2} \)
71 \( 1 + 1.19e4T + 1.80e9T^{2} \)
73 \( 1 - 2.50e4T + 2.07e9T^{2} \)
79 \( 1 - 1.78e4T + 3.07e9T^{2} \)
83 \( 1 + 2.76e3T + 3.93e9T^{2} \)
89 \( 1 - 1.79e4T + 5.58e9T^{2} \)
97 \( 1 + 1.66e5T + 8.58e9T^{2} \)
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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

−11.30046118165570337950883991134, −9.895484506633766733194156708892, −9.259552237899023496776905934276, −8.208684894129665812674373902585, −7.61803038500013563900933817621, −6.45715906528672654029424610486, −4.20809623814440702891570597989, −3.45290446205693656438848701835, −1.31209003978485581879843004447, 0, 1.31209003978485581879843004447, 3.45290446205693656438848701835, 4.20809623814440702891570597989, 6.45715906528672654029424610486, 7.61803038500013563900933817621, 8.208684894129665812674373902585, 9.259552237899023496776905934276, 9.895484506633766733194156708892, 11.30046118165570337950883991134

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