Properties

Label 2-177-3.2-c4-0-39
Degree $2$
Conductor $177$
Sign $0.546 + 0.837i$
Analytic cond. $18.2964$
Root an. cond. $4.27743$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.16i·2-s + (−7.53 + 4.91i)3-s + 14.6·4-s + 22.4i·5-s + (5.73 + 8.79i)6-s − 87.0·7-s − 35.7i·8-s + (32.6 − 74.1i)9-s + 26.2·10-s − 176. i·11-s + (−110. + 71.9i)12-s − 62.6·13-s + 101. i·14-s + (−110. − 169. i)15-s + 192.·16-s + 275. i·17-s + ⋯
L(s)  = 1  − 0.291i·2-s + (−0.837 + 0.546i)3-s + 0.914·4-s + 0.898i·5-s + (0.159 + 0.244i)6-s − 1.77·7-s − 0.558i·8-s + (0.403 − 0.915i)9-s + 0.262·10-s − 1.45i·11-s + (−0.766 + 0.499i)12-s − 0.370·13-s + 0.518i·14-s + (−0.490 − 0.752i)15-s + 0.751·16-s + 0.952i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.546 + 0.837i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ 0.546 + 0.837i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.155963761\)
\(L(\frac12)\) \(\approx\) \(1.155963761\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (7.53 - 4.91i)T \)
59 \( 1 + 453. iT \)
good2 \( 1 + 1.16iT - 16T^{2} \)
5 \( 1 - 22.4iT - 625T^{2} \)
7 \( 1 + 87.0T + 2.40e3T^{2} \)
11 \( 1 + 176. iT - 1.46e4T^{2} \)
13 \( 1 + 62.6T + 2.85e4T^{2} \)
17 \( 1 - 275. iT - 8.35e4T^{2} \)
19 \( 1 - 639.T + 1.30e5T^{2} \)
23 \( 1 + 459. iT - 2.79e5T^{2} \)
29 \( 1 + 559. iT - 7.07e5T^{2} \)
31 \( 1 - 111.T + 9.23e5T^{2} \)
37 \( 1 + 238.T + 1.87e6T^{2} \)
41 \( 1 + 1.87e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.85e3T + 3.41e6T^{2} \)
47 \( 1 + 2.04e3iT - 4.87e6T^{2} \)
53 \( 1 - 243. iT - 7.89e6T^{2} \)
61 \( 1 + 4.62e3T + 1.38e7T^{2} \)
67 \( 1 + 5.99e3T + 2.01e7T^{2} \)
71 \( 1 + 8.08e3iT - 2.54e7T^{2} \)
73 \( 1 - 321.T + 2.83e7T^{2} \)
79 \( 1 + 105.T + 3.89e7T^{2} \)
83 \( 1 + 1.24e4iT - 4.74e7T^{2} \)
89 \( 1 - 4.81e3iT - 6.27e7T^{2} \)
97 \( 1 + 6.90e3T + 8.85e7T^{2} \)
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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

−11.75871758256457262371673346278, −10.67695939000971641098007345477, −10.30575448688727603305345699907, −9.226093721997425721244824626629, −7.29434383552799632076446249047, −6.34895548907543289914098822276, −5.84686231314198545809346552846, −3.57604356692320570208254858229, −2.96351632808406713949819770984, −0.52533176844496478129809495238, 1.18466109716623258739162738600, 2.85446279815495034928722910070, 4.91116526151949797156088223929, 5.89757862007293145821035250580, 7.06764068831627861234813985010, 7.44664581188413660116841460533, 9.400775113906133794540181981242, 10.04519564775082840977293917105, 11.48021428438643367938882563199, 12.34978404848881344126174545444

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