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Properties

Label 2-177-59.58-c8-0-23
Degree $2$
Conductor $177$
Sign $0.709 + 0.704i$
Analytic cond. $72.1060$
Root an. cond. $8.49152$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 28.5i·2-s − 46.7·3-s − 558.·4-s + 1.01e3·5-s + 1.33e3i·6-s + 1.13e3·7-s + 8.63e3i·8-s + 2.18e3·9-s − 2.90e4i·10-s + 4.60e3i·11-s + 2.61e4·12-s − 8.98e3i·13-s − 3.23e4i·14-s − 4.76e4·15-s + 1.03e5·16-s − 1.41e5·17-s + ⋯
L(s)  = 1  − 1.78i·2-s − 0.577·3-s − 2.18·4-s + 1.63·5-s + 1.02i·6-s + 0.471·7-s + 2.10i·8-s + 0.333·9-s − 2.90i·10-s + 0.314i·11-s + 1.25·12-s − 0.314i·13-s − 0.841i·14-s − 0.941·15-s + 1.57·16-s − 1.69·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.709 + 0.704i$
Analytic conductor: \(72.1060\)
Root analytic conductor: \(8.49152\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :4),\ 0.709 + 0.704i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.641404329\)
\(L(\frac12)\) \(\approx\) \(1.641404329\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 46.7T \)
59 \( 1 + (-8.59e6 - 8.54e6i)T \)
good2 \( 1 + 28.5iT - 256T^{2} \)
5 \( 1 - 1.01e3T + 3.90e5T^{2} \)
7 \( 1 - 1.13e3T + 5.76e6T^{2} \)
11 \( 1 - 4.60e3iT - 2.14e8T^{2} \)
13 \( 1 + 8.98e3iT - 8.15e8T^{2} \)
17 \( 1 + 1.41e5T + 6.97e9T^{2} \)
19 \( 1 - 3.76e4T + 1.69e10T^{2} \)
23 \( 1 - 3.68e5iT - 7.83e10T^{2} \)
29 \( 1 - 1.07e6T + 5.00e11T^{2} \)
31 \( 1 - 1.36e6iT - 8.52e11T^{2} \)
37 \( 1 - 4.54e5iT - 3.51e12T^{2} \)
41 \( 1 + 2.19e6T + 7.98e12T^{2} \)
43 \( 1 + 1.55e5iT - 1.16e13T^{2} \)
47 \( 1 - 4.12e6iT - 2.38e13T^{2} \)
53 \( 1 + 8.47e6T + 6.22e13T^{2} \)
61 \( 1 - 1.90e7iT - 1.91e14T^{2} \)
67 \( 1 - 1.59e6iT - 4.06e14T^{2} \)
71 \( 1 - 2.82e7T + 6.45e14T^{2} \)
73 \( 1 - 3.05e7iT - 8.06e14T^{2} \)
79 \( 1 + 4.78e6T + 1.51e15T^{2} \)
83 \( 1 + 6.63e7iT - 2.25e15T^{2} \)
89 \( 1 - 8.43e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.27e8iT - 7.83e15T^{2} \)
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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

−10.99482662440332173349155039762, −10.27369935334646607742810355963, −9.552289426209173097324379435322, −8.636583183716638600517293933698, −6.66462612543536770745428842356, −5.33799002019191756337567090073, −4.55880278151869900664965837558, −2.91002530025606321028963107834, −1.85353278427275945660508517858, −1.15758732356679989020506755928, 0.44658913524602424945078485976, 2.10010793555203110045796696538, 4.52803155685467715224513290205, 5.23914174316657023961213321511, 6.41717304058895547863242043449, 6.57728720884731764144290603627, 8.184892007465084844883688969805, 9.072416248007431326771214203589, 9.977778620503091194397984519151, 11.15333325792221250855925002489

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