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Properties

Label 2-177-59.22-c1-0-3
Degree $2$
Conductor $177$
Sign $0.590 - 0.806i$
Analytic cond. $1.41335$
Root an. cond. $1.18884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0194 + 0.118i)2-s + (0.267 + 0.963i)3-s + (1.88 − 0.633i)4-s + (−1.57 + 2.32i)5-s + (−0.109 + 0.0504i)6-s + (1.03 + 0.228i)7-s + (0.224 + 0.423i)8-s + (−0.856 + 0.515i)9-s + (−0.306 − 0.142i)10-s + (−0.225 − 0.171i)11-s + (1.11 + 1.64i)12-s + (0.273 + 0.164i)13-s + (−0.00692 + 0.127i)14-s + (−2.66 − 0.898i)15-s + (3.11 − 2.36i)16-s + (0.728 − 0.160i)17-s + ⋯
L(s)  = 1  + (0.0137 + 0.0839i)2-s + (0.154 + 0.556i)3-s + (0.940 − 0.316i)4-s + (−0.705 + 1.04i)5-s + (−0.0445 + 0.0206i)6-s + (0.392 + 0.0863i)7-s + (0.0793 + 0.149i)8-s + (−0.285 + 0.171i)9-s + (−0.0970 − 0.0449i)10-s + (−0.0680 − 0.0517i)11-s + (0.321 + 0.474i)12-s + (0.0759 + 0.0457i)13-s + (−0.00184 + 0.0341i)14-s + (−0.688 − 0.231i)15-s + (0.778 − 0.592i)16-s + (0.176 − 0.0388i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.590 - 0.806i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.590 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20467 + 0.610859i\)
\(L(\frac12)\) \(\approx\) \(1.20467 + 0.610859i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.267 - 0.963i)T \)
59 \( 1 + (-4.51 - 6.21i)T \)
good2 \( 1 + (-0.0194 - 0.118i)T + (-1.89 + 0.638i)T^{2} \)
5 \( 1 + (1.57 - 2.32i)T + (-1.85 - 4.64i)T^{2} \)
7 \( 1 + (-1.03 - 0.228i)T + (6.35 + 2.93i)T^{2} \)
11 \( 1 + (0.225 + 0.171i)T + (2.94 + 10.5i)T^{2} \)
13 \( 1 + (-0.273 - 0.164i)T + (6.08 + 11.4i)T^{2} \)
17 \( 1 + (-0.728 + 0.160i)T + (15.4 - 7.13i)T^{2} \)
19 \( 1 + (2.09 + 0.228i)T + (18.5 + 4.08i)T^{2} \)
23 \( 1 + (-4.27 + 5.03i)T + (-3.72 - 22.6i)T^{2} \)
29 \( 1 + (-0.142 + 0.870i)T + (-27.4 - 9.25i)T^{2} \)
31 \( 1 + (8.02 - 0.872i)T + (30.2 - 6.66i)T^{2} \)
37 \( 1 + (-2.76 + 5.21i)T + (-20.7 - 30.6i)T^{2} \)
41 \( 1 + (5.00 + 5.88i)T + (-6.63 + 40.4i)T^{2} \)
43 \( 1 + (-7.12 + 5.41i)T + (11.5 - 41.4i)T^{2} \)
47 \( 1 + (4.51 + 6.65i)T + (-17.3 + 43.6i)T^{2} \)
53 \( 1 + (7.81 - 3.61i)T + (34.3 - 40.3i)T^{2} \)
61 \( 1 + (0.0621 + 0.379i)T + (-57.8 + 19.4i)T^{2} \)
67 \( 1 + (-5.29 - 9.98i)T + (-37.5 + 55.4i)T^{2} \)
71 \( 1 + (-3.19 - 4.71i)T + (-26.2 + 65.9i)T^{2} \)
73 \( 1 + (0.658 - 12.1i)T + (-72.5 - 7.89i)T^{2} \)
79 \( 1 + (2.62 - 9.43i)T + (-67.6 - 40.7i)T^{2} \)
83 \( 1 + (-5.69 + 5.39i)T + (4.49 - 82.8i)T^{2} \)
89 \( 1 + (0.335 - 2.04i)T + (-84.3 - 28.4i)T^{2} \)
97 \( 1 + (0.218 + 4.03i)T + (-96.4 + 10.4i)T^{2} \)
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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

−12.66614853936593133095501489503, −11.44995778078783809539466762136, −10.94861728764431283083636165014, −10.20375554975404599233594963068, −8.728119170282564657768365332418, −7.51216850762547093770137604045, −6.71080965802670179262337237021, −5.36525402563054752531442907929, −3.74895070048317998626919465500, −2.47000565749625422618449060306, 1.53963738354308464159027612965, 3.34372053248028078771039780752, 4.87212624481663295358918052588, 6.35613093467699599730228148070, 7.62243741568081146429905441493, 8.137508073041888989192393265450, 9.356610144325041572349221078711, 10.99014098960588802969236921867, 11.60264032404904257591071012061, 12.60713057461710340497557590587

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