Properties

Label 2-177-3.2-c2-0-8
Degree $2$
Conductor $177$
Sign $0.749 + 0.662i$
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  − 3.62i·2-s + (−1.98 + 2.24i)3-s − 9.11·4-s + 2.81i·5-s + (8.14 + 7.19i)6-s + 8.99·7-s + 18.5i·8-s + (−1.10 − 8.93i)9-s + 10.1·10-s + 19.0i·11-s + (18.0 − 20.4i)12-s + 8.47·13-s − 32.5i·14-s + (−6.32 − 5.58i)15-s + 30.5·16-s − 19.9i·17-s + ⋯
L(s)  = 1  − 1.81i·2-s + (−0.662 + 0.749i)3-s − 2.27·4-s + 0.562i·5-s + (1.35 + 1.19i)6-s + 1.28·7-s + 2.31i·8-s + (−0.123 − 0.992i)9-s + 1.01·10-s + 1.72i·11-s + (1.50 − 1.70i)12-s + 0.651·13-s − 2.32i·14-s + (−0.421 − 0.372i)15-s + 1.91·16-s − 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.749 + 0.662i$
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: \((2,\ 177,\ (\ :1),\ 0.749 + 0.662i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.08256 - 0.409723i\)
\(L(\frac12)\) \(\approx\) \(1.08256 - 0.409723i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.98 - 2.24i)T \)
59 \( 1 - 7.68iT \)
good2 \( 1 + 3.62iT - 4T^{2} \)
5 \( 1 - 2.81iT - 25T^{2} \)
7 \( 1 - 8.99T + 49T^{2} \)
11 \( 1 - 19.0iT - 121T^{2} \)
13 \( 1 - 8.47T + 169T^{2} \)
17 \( 1 + 19.9iT - 289T^{2} \)
19 \( 1 - 16.3T + 361T^{2} \)
23 \( 1 - 10.3iT - 529T^{2} \)
29 \( 1 - 47.7iT - 841T^{2} \)
31 \( 1 - 4.05T + 961T^{2} \)
37 \( 1 + 13.8T + 1.36e3T^{2} \)
41 \( 1 - 55.0iT - 1.68e3T^{2} \)
43 \( 1 - 44.9T + 1.84e3T^{2} \)
47 \( 1 - 2.46iT - 2.20e3T^{2} \)
53 \( 1 + 74.2iT - 2.80e3T^{2} \)
61 \( 1 - 32.9T + 3.72e3T^{2} \)
67 \( 1 + 51.8T + 4.48e3T^{2} \)
71 \( 1 + 95.1iT - 5.04e3T^{2} \)
73 \( 1 + 123.T + 5.32e3T^{2} \)
79 \( 1 + 138.T + 6.24e3T^{2} \)
83 \( 1 - 27.4iT - 6.88e3T^{2} \)
89 \( 1 + 89.1iT - 7.92e3T^{2} \)
97 \( 1 - 17.0T + 9.40e3T^{2} \)
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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

−11.85630327039152000334254765052, −11.38123510579792995050983039331, −10.57420043033677905175045887461, −9.799868513570066424014090716402, −8.902401667476516055507428041402, −7.22972528460054163776443977003, −5.12279379075098095986690619705, −4.51214358556778799156511663961, −3.10640193621168284798465060449, −1.44896320266598951591141914237, 0.927585529922341034847976401248, 4.36514091712437406610423595112, 5.60246925380425717851985227967, 6.01746149899666844655128816930, 7.40879976094553884682562667197, 8.336374623078532211546375275701, 8.682244899461100370010383740275, 10.69494781256947025199233787909, 11.68398484726026863880142419882, 12.97933098427639563609329308884

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