Properties

Label 2-177-3.2-c2-0-9
Degree $2$
Conductor $177$
Sign $0.705 - 0.709i$
Analytic cond. $4.82290$
Root an. cond. $2.19611$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.201i·2-s + (−2.12 − 2.11i)3-s + 3.95·4-s + 6.21i·5-s + (0.425 − 0.428i)6-s − 0.661·7-s + 1.60i·8-s + (0.0531 + 8.99i)9-s − 1.25·10-s + 5.79i·11-s + (−8.42 − 8.37i)12-s + 3.06·13-s − 0.133i·14-s + (13.1 − 13.2i)15-s + 15.5·16-s + 12.8i·17-s + ⋯
L(s)  = 1  + 0.100i·2-s + (−0.709 − 0.705i)3-s + 0.989·4-s + 1.24i·5-s + (0.0709 − 0.0714i)6-s − 0.0945·7-s + 0.200i·8-s + (0.00590 + 0.999i)9-s − 0.125·10-s + 0.526i·11-s + (−0.702 − 0.697i)12-s + 0.235·13-s − 0.00951i·14-s + (0.876 − 0.881i)15-s + 0.969·16-s + 0.754i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 - 0.709i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.705 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.705 - 0.709i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ 0.705 - 0.709i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.33232 + 0.554176i\)
\(L(\frac12)\) \(\approx\) \(1.33232 + 0.554176i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.12 + 2.11i)T \)
59 \( 1 + 7.68iT \)
good2 \( 1 - 0.201iT - 4T^{2} \)
5 \( 1 - 6.21iT - 25T^{2} \)
7 \( 1 + 0.661T + 49T^{2} \)
11 \( 1 - 5.79iT - 121T^{2} \)
13 \( 1 - 3.06T + 169T^{2} \)
17 \( 1 - 12.8iT - 289T^{2} \)
19 \( 1 - 24.1T + 361T^{2} \)
23 \( 1 - 10.5iT - 529T^{2} \)
29 \( 1 + 3.55iT - 841T^{2} \)
31 \( 1 - 23.2T + 961T^{2} \)
37 \( 1 + 63.0T + 1.36e3T^{2} \)
41 \( 1 + 25.4iT - 1.68e3T^{2} \)
43 \( 1 + 26.9T + 1.84e3T^{2} \)
47 \( 1 + 4.63iT - 2.20e3T^{2} \)
53 \( 1 + 39.5iT - 2.80e3T^{2} \)
61 \( 1 - 48.9T + 3.72e3T^{2} \)
67 \( 1 - 35.5T + 4.48e3T^{2} \)
71 \( 1 + 103. iT - 5.04e3T^{2} \)
73 \( 1 + 76.1T + 5.32e3T^{2} \)
79 \( 1 - 26.3T + 6.24e3T^{2} \)
83 \( 1 - 52.3iT - 6.88e3T^{2} \)
89 \( 1 + 7.05iT - 7.92e3T^{2} \)
97 \( 1 - 106.T + 9.40e3T^{2} \)
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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

−12.29328235239802162329307422403, −11.59227451627383686453106119589, −10.76828434222080281877915396580, −10.03810495676138561051533851683, −8.026378761693394569186010551621, −7.07926376164753467412226107989, −6.53964483585477533711770753257, −5.43002109944633719173205908199, −3.23046664924610284579866114528, −1.82133539160485636826796782056, 1.01783959824420087038746807049, 3.28309049174242978690687048305, 4.83888187783805850488594155057, 5.74339873526850795257067214963, 6.93195238621411098886046424789, 8.391542051738397213019661207078, 9.474091445370993579568444557585, 10.42268671565089812268866145147, 11.54153463881595870153338680640, 12.00162295417933949014040747372

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