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Properties

Label 2-177-177.176-c5-0-83
Degree $2$
Conductor $177$
Sign $-0.353 - 0.935i$
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  − 8.73·2-s + (1.48 − 15.5i)3-s + 44.2·4-s − 95.2i·5-s + (−12.9 + 135. i)6-s + 144.·7-s − 106.·8-s + (−238. − 46.0i)9-s + 831. i·10-s − 217.·11-s + (65.6 − 686. i)12-s − 156. i·13-s − 1.26e3·14-s + (−1.47e3 − 141. i)15-s − 482.·16-s + 457. i·17-s + ⋯
L(s)  = 1  − 1.54·2-s + (0.0952 − 0.995i)3-s + 1.38·4-s − 1.70i·5-s + (−0.146 + 1.53i)6-s + 1.11·7-s − 0.590·8-s + (−0.981 − 0.189i)9-s + 2.63i·10-s − 0.542·11-s + (0.131 − 1.37i)12-s − 0.256i·13-s − 1.71·14-s + (−1.69 − 0.162i)15-s − 0.471·16-s + 0.384i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.353 - 0.935i$
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: \((2,\ 177,\ (\ :5/2),\ -0.353 - 0.935i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.3862154357\)
\(L(\frac12)\) \(\approx\) \(0.3862154357\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.48 + 15.5i)T \)
59 \( 1 + (2.39e4 - 1.17e4i)T \)
good2 \( 1 + 8.73T + 32T^{2} \)
5 \( 1 + 95.2iT - 3.12e3T^{2} \)
7 \( 1 - 144.T + 1.68e4T^{2} \)
11 \( 1 + 217.T + 1.61e5T^{2} \)
13 \( 1 + 156. iT - 3.71e5T^{2} \)
17 \( 1 - 457. iT - 1.41e6T^{2} \)
19 \( 1 - 309.T + 2.47e6T^{2} \)
23 \( 1 + 1.50e3T + 6.43e6T^{2} \)
29 \( 1 + 3.62e3iT - 2.05e7T^{2} \)
31 \( 1 + 6.83e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.68e3iT - 6.93e7T^{2} \)
41 \( 1 + 7.53e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.72e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.84e4T + 2.29e8T^{2} \)
53 \( 1 - 6.02e3iT - 4.18e8T^{2} \)
61 \( 1 - 4.09e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.55e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.85e4iT - 1.80e9T^{2} \)
73 \( 1 + 4.52e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.36e4T + 3.07e9T^{2} \)
83 \( 1 + 3.10e4T + 3.93e9T^{2} \)
89 \( 1 + 1.19e4T + 5.58e9T^{2} \)
97 \( 1 - 8.52e4iT - 8.58e9T^{2} \)
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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

−11.14489302154151758742250460361, −9.758911532058043045357920697942, −8.783759838662161053854599702358, −8.029908185045767197308777030284, −7.71949148576479092310194589005, −5.94451004578165149888219881936, −4.68034348345653816788586065439, −2.04950822927873180002976658971, −1.20459587054372432403834806940, −0.21745465562728098063115680980, 1.99131506155561100408475683668, 3.23985640863653426219725812109, 4.96042363351919582155149364157, 6.61212613483428420121844231011, 7.67378163080409825438738598473, 8.478015811095334998279439406619, 9.675654874211406236074287766913, 10.42977651246757961232570502398, 11.00805269136132222662501099568, 11.64413482069561111999096200277

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