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Properties

Label 2-177-59.58-c10-0-30
Degree $2$
Conductor $177$
Sign $-0.949 + 0.315i$
Analytic cond. $112.458$
Root an. cond. $10.6046$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 50.8i·2-s − 140.·3-s − 1.55e3·4-s + 737.·5-s + 7.12e3i·6-s − 2.12e4·7-s + 2.70e4i·8-s + 1.96e4·9-s − 3.74e4i·10-s + 2.65e5i·11-s + 2.18e5·12-s + 4.70e5i·13-s + 1.08e6i·14-s − 1.03e5·15-s − 2.18e5·16-s − 1.88e6·17-s + ⋯
L(s)  = 1  − 1.58i·2-s − 0.577·3-s − 1.52·4-s + 0.235·5-s + 0.916i·6-s − 1.26·7-s + 0.826i·8-s + 0.333·9-s − 0.374i·10-s + 1.64i·11-s + 0.877·12-s + 1.26i·13-s + 2.01i·14-s − 0.136·15-s − 0.208·16-s − 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.315i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.949 + 0.315i$
Analytic conductor: \(112.458\)
Root analytic conductor: \(10.6046\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5),\ -0.949 + 0.315i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.1632290352\)
\(L(\frac12)\) \(\approx\) \(0.1632290352\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 140.T \)
59 \( 1 + (-6.78e8 + 2.25e8i)T \)
good2 \( 1 + 50.8iT - 1.02e3T^{2} \)
5 \( 1 - 737.T + 9.76e6T^{2} \)
7 \( 1 + 2.12e4T + 2.82e8T^{2} \)
11 \( 1 - 2.65e5iT - 2.59e10T^{2} \)
13 \( 1 - 4.70e5iT - 1.37e11T^{2} \)
17 \( 1 + 1.88e6T + 2.01e12T^{2} \)
19 \( 1 + 4.41e6T + 6.13e12T^{2} \)
23 \( 1 - 1.07e7iT - 4.14e13T^{2} \)
29 \( 1 - 1.25e7T + 4.20e14T^{2} \)
31 \( 1 + 4.28e7iT - 8.19e14T^{2} \)
37 \( 1 - 2.69e7iT - 4.80e15T^{2} \)
41 \( 1 - 4.52e6T + 1.34e16T^{2} \)
43 \( 1 + 1.84e8iT - 2.16e16T^{2} \)
47 \( 1 - 1.36e8iT - 5.25e16T^{2} \)
53 \( 1 + 4.28e7T + 1.74e17T^{2} \)
61 \( 1 - 4.89e8iT - 7.13e17T^{2} \)
67 \( 1 - 8.12e8iT - 1.82e18T^{2} \)
71 \( 1 + 2.09e9T + 3.25e18T^{2} \)
73 \( 1 + 3.96e9iT - 4.29e18T^{2} \)
79 \( 1 + 2.20e9T + 9.46e18T^{2} \)
83 \( 1 + 2.68e9iT - 1.55e19T^{2} \)
89 \( 1 + 3.26e9iT - 3.11e19T^{2} \)
97 \( 1 - 1.33e10iT - 7.37e19T^{2} \)
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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

−10.39714604544678093252532977899, −9.680557160865072835320204045488, −9.096857094734014899202528184250, −7.06951487115232819632636909570, −6.24766254463456320516602941197, −4.50254330987199618874257357542, −3.93443481042073946040722933132, −2.30440628781951249901916030129, −1.76938427066105928116131178700, −0.081987831717783314936628006252, 0.42621637780897782068648125456, 2.74617005324111923839436817344, 4.22249105474011037283378983896, 5.50782421655507487583158839652, 6.36172902650745645935056583503, 6.64408673005328727268031372933, 8.281258430852002457256542460966, 8.797654926446970933625099040155, 10.23009816336234829836161807821, 11.04227666398933479805747152015

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