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Properties

Label 2-177-177.176-c5-0-33
Degree $2$
Conductor $177$
Sign $0.732 - 0.680i$
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  − 7.43·2-s + (−12.1 − 9.70i)3-s + 23.2·4-s + 26.1i·5-s + (90.6 + 72.1i)6-s + 171.·7-s + 64.8·8-s + (54.4 + 236. i)9-s − 194. i·10-s + 440.·11-s + (−283. − 226. i)12-s − 280. i·13-s − 1.27e3·14-s + (253. − 318. i)15-s − 1.22e3·16-s + 1.40e3i·17-s + ⋯
L(s)  = 1  − 1.31·2-s + (−0.782 − 0.622i)3-s + 0.727·4-s + 0.466i·5-s + (1.02 + 0.818i)6-s + 1.32·7-s + 0.358·8-s + (0.224 + 0.974i)9-s − 0.613i·10-s + 1.09·11-s + (−0.569 − 0.453i)12-s − 0.459i·13-s − 1.74·14-s + (0.290 − 0.365i)15-s − 1.19·16-s + 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.732 - 0.680i$
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: \((2,\ 177,\ (\ :5/2),\ 0.732 - 0.680i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7934702525\)
\(L(\frac12)\) \(\approx\) \(0.7934702525\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (12.1 + 9.70i)T \)
59 \( 1 + (-3.99e3 + 2.64e4i)T \)
good2 \( 1 + 7.43T + 32T^{2} \)
5 \( 1 - 26.1iT - 3.12e3T^{2} \)
7 \( 1 - 171.T + 1.68e4T^{2} \)
11 \( 1 - 440.T + 1.61e5T^{2} \)
13 \( 1 + 280. iT - 3.71e5T^{2} \)
17 \( 1 - 1.40e3iT - 1.41e6T^{2} \)
19 \( 1 - 616.T + 2.47e6T^{2} \)
23 \( 1 + 4.16e3T + 6.43e6T^{2} \)
29 \( 1 - 6.60e3iT - 2.05e7T^{2} \)
31 \( 1 + 5.98e3iT - 2.86e7T^{2} \)
37 \( 1 + 2.03e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.97e3iT - 1.15e8T^{2} \)
43 \( 1 + 2.46e3iT - 1.47e8T^{2} \)
47 \( 1 - 4.98e3T + 2.29e8T^{2} \)
53 \( 1 - 3.61e4iT - 4.18e8T^{2} \)
61 \( 1 - 76.2iT - 8.44e8T^{2} \)
67 \( 1 - 4.37e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.73e3iT - 1.80e9T^{2} \)
73 \( 1 - 7.97e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.24e4T + 3.07e9T^{2} \)
83 \( 1 + 1.28e4T + 3.93e9T^{2} \)
89 \( 1 - 9.17e4T + 5.58e9T^{2} \)
97 \( 1 + 1.24e5iT - 8.58e9T^{2} \)
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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

−11.52223072748572100261824194924, −10.88734540648520348569959214148, −10.08127993460998102419095920551, −8.665063869118431454676705374115, −7.892432270786767469580904290401, −7.02005480151725057851949081491, −5.81280014874387582423574150689, −4.34373960652504640073886473243, −1.89746647022293341228546995402, −1.05530878618793275894496533956, 0.57372748345515097997349037016, 1.64309572596681227602180268240, 4.22534142123870209956575232150, 5.06304730246940167044297757811, 6.60634417248182839386417762827, 7.82913596822604610728667188056, 8.869707092725603823682485348808, 9.567193153564092324963796156710, 10.53354983006462817767970297612, 11.63348216241253035811986083729

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