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Properties

Label 2-177-59.58-c8-0-0
Degree $2$
Conductor $177$
Sign $0.882 - 0.470i$
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  − 27.0i·2-s + 46.7·3-s − 474.·4-s + 21.6·5-s − 1.26e3i·6-s − 141.·7-s + 5.91e3i·8-s + 2.18e3·9-s − 585. i·10-s + 7.99e3i·11-s − 2.22e4·12-s − 5.52e4i·13-s + 3.82e3i·14-s + 1.01e3·15-s + 3.84e4·16-s + 1.24e4·17-s + ⋯
L(s)  = 1  − 1.68i·2-s + 0.577·3-s − 1.85·4-s + 0.0346·5-s − 0.975i·6-s − 0.0589·7-s + 1.44i·8-s + 0.333·9-s − 0.0585i·10-s + 0.546i·11-s − 1.07·12-s − 1.93i·13-s + 0.0996i·14-s + 0.0200·15-s + 0.586·16-s + 0.148·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.882 - 0.470i$
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: \((2,\ 177,\ (\ :4),\ 0.882 - 0.470i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.2530501320\)
\(L(\frac12)\) \(\approx\) \(0.2530501320\)
\(L(5)\) 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 \( 1 + (1.06e7 - 5.70e6i)T \)
good2 \( 1 + 27.0iT - 256T^{2} \)
5 \( 1 - 21.6T + 3.90e5T^{2} \)
7 \( 1 + 141.T + 5.76e6T^{2} \)
11 \( 1 - 7.99e3iT - 2.14e8T^{2} \)
13 \( 1 + 5.52e4iT - 8.15e8T^{2} \)
17 \( 1 - 1.24e4T + 6.97e9T^{2} \)
19 \( 1 + 1.30e5T + 1.69e10T^{2} \)
23 \( 1 - 3.79e5iT - 7.83e10T^{2} \)
29 \( 1 + 1.49e5T + 5.00e11T^{2} \)
31 \( 1 + 1.11e6iT - 8.52e11T^{2} \)
37 \( 1 - 2.51e6iT - 3.51e12T^{2} \)
41 \( 1 - 1.71e5T + 7.98e12T^{2} \)
43 \( 1 + 3.98e6iT - 1.16e13T^{2} \)
47 \( 1 - 5.38e6iT - 2.38e13T^{2} \)
53 \( 1 - 6.71e6T + 6.22e13T^{2} \)
61 \( 1 - 2.35e7iT - 1.91e14T^{2} \)
67 \( 1 - 3.32e7iT - 4.06e14T^{2} \)
71 \( 1 - 2.18e7T + 6.45e14T^{2} \)
73 \( 1 + 5.20e6iT - 8.06e14T^{2} \)
79 \( 1 + 3.31e7T + 1.51e15T^{2} \)
83 \( 1 - 3.11e7iT - 2.25e15T^{2} \)
89 \( 1 + 8.38e7iT - 3.93e15T^{2} \)
97 \( 1 - 2.10e7iT - 7.83e15T^{2} \)
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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

−11.24549188687277514341270544800, −10.19971041183024076758735374316, −9.739469352927351194658490927808, −8.535745668618457879902573409060, −7.54928256156619837837507513796, −5.65723789871217698073156242573, −4.27644409986933826728972578266, −3.28561990183080521345530411864, −2.35754961857846423587066536416, −1.23259133267719354396926866629, 0.05576726842151713749762741724, 2.01958921208434202986496936310, 3.88133627965326461799624640223, 4.82322770742416470247102886489, 6.26040076673409806975188789430, 6.83620993209788764010631595083, 8.022636435160846278333160142696, 8.794493210830335737828530202465, 9.528393599401692117319883826982, 11.01925996570925985465669475014

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