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Properties

Label 2-177-177.176-c5-0-1
Degree $2$
Conductor $177$
Sign $0.0374 + 0.999i$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.97·2-s + (12.6 + 9.05i)3-s − 7.29·4-s + 63.5i·5-s + (−63.0 − 45.0i)6-s − 191.·7-s + 195.·8-s + (78.9 + 229. i)9-s − 315. i·10-s − 473.·11-s + (−92.5 − 66.0i)12-s + 634. i·13-s + 952.·14-s + (−575. + 806. i)15-s − 737.·16-s − 1.37e3i·17-s + ⋯
L(s)  = 1  − 0.878·2-s + (0.813 + 0.580i)3-s − 0.227·4-s + 1.13i·5-s + (−0.715 − 0.510i)6-s − 1.47·7-s + 1.07·8-s + (0.325 + 0.945i)9-s − 0.998i·10-s − 1.17·11-s + (−0.185 − 0.132i)12-s + 1.04i·13-s + 1.29·14-s + (−0.660 + 0.925i)15-s − 0.720·16-s − 1.15i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0374 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0374 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.0374 + 0.999i$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ 0.0374 + 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.02582005877\)
\(L(\frac12)\) \(\approx\) \(0.02582005877\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-12.6 - 9.05i)T \)
59 \( 1 + (1.63e4 + 2.11e4i)T \)
good2 \( 1 + 4.97T + 32T^{2} \)
5 \( 1 - 63.5iT - 3.12e3T^{2} \)
7 \( 1 + 191.T + 1.68e4T^{2} \)
11 \( 1 + 473.T + 1.61e5T^{2} \)
13 \( 1 - 634. iT - 3.71e5T^{2} \)
17 \( 1 + 1.37e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.06e3T + 2.47e6T^{2} \)
23 \( 1 - 3.70e3T + 6.43e6T^{2} \)
29 \( 1 - 2.95e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.13e3iT - 2.86e7T^{2} \)
37 \( 1 + 6.26e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.20e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.48e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.24e4T + 2.29e8T^{2} \)
53 \( 1 - 3.12e3iT - 4.18e8T^{2} \)
61 \( 1 + 3.55e4iT - 8.44e8T^{2} \)
67 \( 1 + 8.88e3iT - 1.35e9T^{2} \)
71 \( 1 + 3.83e4iT - 1.80e9T^{2} \)
73 \( 1 + 7.98e4iT - 2.07e9T^{2} \)
79 \( 1 + 9.89e4T + 3.07e9T^{2} \)
83 \( 1 + 1.06e4T + 3.93e9T^{2} \)
89 \( 1 - 3.78e4T + 5.58e9T^{2} \)
97 \( 1 - 1.64e5iT - 8.58e9T^{2} \)
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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

−12.80852746640057462093640262609, −10.83399467916119187776989550560, −10.44336212973155710791695734654, −9.411332335023222345613117760460, −8.885266961353713997974435329836, −7.47781272913064937663410799067, −6.74538980734760609922472392907, −4.82212355936129549781457128735, −3.38725072942667191397063283883, −2.42276856259915288445297853898, 0.01207808583903291108957565879, 1.01016931650633947793400599857, 2.70761027452040060830057492392, 4.20802130794693459070167102184, 5.79890227816062677739416321424, 7.24896785246027003451562368666, 8.338510253430284559215484472005, 8.774291089921868123627738687832, 9.807197163155715424966579983730, 10.54932646843888702431922502118

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