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Properties

Label 2-177-3.2-c2-0-19
Degree $2$
Conductor $177$
Sign $-0.193 - 0.981i$
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  + 3.18i·2-s + (2.94 − 0.581i)3-s − 6.16·4-s − 2.91i·5-s + (1.85 + 9.38i)6-s + 12.1·7-s − 6.88i·8-s + (8.32 − 3.42i)9-s + 9.30·10-s + 13.3i·11-s + (−18.1 + 3.58i)12-s − 12.2·13-s + 38.8i·14-s + (−1.69 − 8.58i)15-s − 2.68·16-s + 2.10i·17-s + ⋯
L(s)  = 1  + 1.59i·2-s + (0.981 − 0.193i)3-s − 1.54·4-s − 0.583i·5-s + (0.308 + 1.56i)6-s + 1.74·7-s − 0.860i·8-s + (0.924 − 0.380i)9-s + 0.930·10-s + 1.21i·11-s + (−1.51 + 0.298i)12-s − 0.939·13-s + 2.77i·14-s + (−0.113 − 0.572i)15-s − 0.168·16-s + 0.123i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.193 - 0.981i$
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: \((2,\ 177,\ (\ :1),\ -0.193 - 0.981i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.37006 + 1.66730i\)
\(L(\frac12)\) \(\approx\) \(1.37006 + 1.66730i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.94 + 0.581i)T \)
59 \( 1 - 7.68iT \)
good2 \( 1 - 3.18iT - 4T^{2} \)
5 \( 1 + 2.91iT - 25T^{2} \)
7 \( 1 - 12.1T + 49T^{2} \)
11 \( 1 - 13.3iT - 121T^{2} \)
13 \( 1 + 12.2T + 169T^{2} \)
17 \( 1 - 2.10iT - 289T^{2} \)
19 \( 1 + 3.44T + 361T^{2} \)
23 \( 1 + 44.1iT - 529T^{2} \)
29 \( 1 - 41.6iT - 841T^{2} \)
31 \( 1 + 28.5T + 961T^{2} \)
37 \( 1 + 16.7T + 1.36e3T^{2} \)
41 \( 1 + 7.60iT - 1.68e3T^{2} \)
43 \( 1 + 46.0T + 1.84e3T^{2} \)
47 \( 1 - 1.11iT - 2.20e3T^{2} \)
53 \( 1 + 30.2iT - 2.80e3T^{2} \)
61 \( 1 + 46.1T + 3.72e3T^{2} \)
67 \( 1 - 66.4T + 4.48e3T^{2} \)
71 \( 1 + 99.7iT - 5.04e3T^{2} \)
73 \( 1 + 129.T + 5.32e3T^{2} \)
79 \( 1 - 12.2T + 6.24e3T^{2} \)
83 \( 1 + 41.1iT - 6.88e3T^{2} \)
89 \( 1 + 75.0iT - 7.92e3T^{2} \)
97 \( 1 - 186.T + 9.40e3T^{2} \)
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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

−12.98430906768865055964207625716, −12.16908288200946975508170798425, −10.46499692290273242172404974027, −9.035018622978788462321036831291, −8.454256313015338807275187187872, −7.58211842909022703833304475567, −6.87375994108300006168271203997, −4.96121630842498373676780581884, −4.59220288623545738047359064372, −1.95365789034734460745301789476, 1.58792567305277514726525812931, 2.74808253030598608378362665091, 3.88976825281058540464307162923, 5.11945559805448807799653509879, 7.41383278075481547947907192918, 8.380650229258332160458514468024, 9.378338640458043638413994611681, 10.40223776463081203267378506159, 11.22094550863131324466549622685, 11.80986452603418945321381866393

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