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Properties

Label 2-177-177.176-c3-0-12
Degree $2$
Conductor $177$
Sign $-0.103 - 0.994i$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.72·2-s + (−4.94 − 1.59i)3-s + 14.2·4-s + 15.2i·5-s + (23.3 + 7.52i)6-s + 9.03·7-s − 29.6·8-s + (21.9 + 15.7i)9-s − 72.0i·10-s − 12.9·11-s + (−70.6 − 22.7i)12-s − 55.4i·13-s − 42.6·14-s + (24.3 − 75.5i)15-s + 25.6·16-s + 10.8i·17-s + ⋯
L(s)  = 1  − 1.66·2-s + (−0.951 − 0.306i)3-s + 1.78·4-s + 1.36i·5-s + (1.58 + 0.511i)6-s + 0.488·7-s − 1.30·8-s + (0.811 + 0.583i)9-s − 2.27i·10-s − 0.356·11-s + (−1.69 − 0.547i)12-s − 1.18i·13-s − 0.814·14-s + (0.418 − 1.30i)15-s + 0.401·16-s + 0.154i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.103 - 0.994i$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ -0.103 - 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.302728 + 0.335999i\)
\(L(\frac12)\) \(\approx\) \(0.302728 + 0.335999i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.94 + 1.59i)T \)
59 \( 1 + (-183. - 414. i)T \)
good2 \( 1 + 4.72T + 8T^{2} \)
5 \( 1 - 15.2iT - 125T^{2} \)
7 \( 1 - 9.03T + 343T^{2} \)
11 \( 1 + 12.9T + 1.33e3T^{2} \)
13 \( 1 + 55.4iT - 2.19e3T^{2} \)
17 \( 1 - 10.8iT - 4.91e3T^{2} \)
19 \( 1 - 126.T + 6.85e3T^{2} \)
23 \( 1 - 21.4T + 1.21e4T^{2} \)
29 \( 1 - 6.52iT - 2.43e4T^{2} \)
31 \( 1 - 223. iT - 2.97e4T^{2} \)
37 \( 1 - 166. iT - 5.06e4T^{2} \)
41 \( 1 - 152. iT - 6.89e4T^{2} \)
43 \( 1 - 122. iT - 7.95e4T^{2} \)
47 \( 1 - 126.T + 1.03e5T^{2} \)
53 \( 1 + 578. iT - 1.48e5T^{2} \)
61 \( 1 - 272. iT - 2.26e5T^{2} \)
67 \( 1 - 957. iT - 3.00e5T^{2} \)
71 \( 1 - 639. iT - 3.57e5T^{2} \)
73 \( 1 + 869. iT - 3.89e5T^{2} \)
79 \( 1 - 668.T + 4.93e5T^{2} \)
83 \( 1 + 1.15e3T + 5.71e5T^{2} \)
89 \( 1 + 505.T + 7.04e5T^{2} \)
97 \( 1 - 1.50e3iT - 9.12e5T^{2} \)
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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

−11.88803557434340532587914484127, −11.13581350771407754719756530464, −10.45540577353681763655451988368, −9.859834620429722343823039384568, −8.205803914001917464408760155237, −7.41507777210340578795775393365, −6.66935134873482893843179182069, −5.35050955815671503099653489406, −2.85881777319794295183556917065, −1.17827035545334927568107827930, 0.48684855546041194098128201038, 1.61562227227403999758136791954, 4.46498563957708968034538678437, 5.57463688163328151554646239571, 7.05415596770765030049045106608, 8.050958837347556632848995953369, 9.240845876600384163764097997094, 9.603173995459838857655904569584, 10.92283990339250350093809706737, 11.62578928943014061083324873619

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