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Properties

Label 2-177-3.2-c2-0-34
Degree $2$
Conductor $177$
Sign $-0.896 + 0.443i$
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  − 0.472i·2-s + (−1.33 − 2.68i)3-s + 3.77·4-s − 7.63i·5-s + (−1.27 + 0.628i)6-s − 9.20·7-s − 3.67i·8-s + (−5.46 + 7.15i)9-s − 3.61·10-s + 2.06i·11-s + (−5.02 − 10.1i)12-s + 15.4·13-s + 4.35i·14-s + (−20.5 + 10.1i)15-s + 13.3·16-s + 15.6i·17-s + ⋯
L(s)  = 1  − 0.236i·2-s + (−0.443 − 0.896i)3-s + 0.944·4-s − 1.52i·5-s + (−0.211 + 0.104i)6-s − 1.31·7-s − 0.459i·8-s + (−0.606 + 0.794i)9-s − 0.361·10-s + 0.187i·11-s + (−0.418 − 0.846i)12-s + 1.18·13-s + 0.310i·14-s + (−1.36 + 0.677i)15-s + 0.835·16-s + 0.921i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.896 + 0.443i$
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: \((2,\ 177,\ (\ :1),\ -0.896 + 0.443i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.276901 - 1.18442i\)
\(L(\frac12)\) \(\approx\) \(0.276901 - 1.18442i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.33 + 2.68i)T \)
59 \( 1 - 7.68iT \)
good2 \( 1 + 0.472iT - 4T^{2} \)
5 \( 1 + 7.63iT - 25T^{2} \)
7 \( 1 + 9.20T + 49T^{2} \)
11 \( 1 - 2.06iT - 121T^{2} \)
13 \( 1 - 15.4T + 169T^{2} \)
17 \( 1 - 15.6iT - 289T^{2} \)
19 \( 1 + 31.9T + 361T^{2} \)
23 \( 1 + 41.9iT - 529T^{2} \)
29 \( 1 + 41.7iT - 841T^{2} \)
31 \( 1 + 6.91T + 961T^{2} \)
37 \( 1 - 63.0T + 1.36e3T^{2} \)
41 \( 1 - 28.7iT - 1.68e3T^{2} \)
43 \( 1 - 59.9T + 1.84e3T^{2} \)
47 \( 1 + 37.1iT - 2.20e3T^{2} \)
53 \( 1 - 0.0188iT - 2.80e3T^{2} \)
61 \( 1 + 21.9T + 3.72e3T^{2} \)
67 \( 1 - 25.9T + 4.48e3T^{2} \)
71 \( 1 + 40.5iT - 5.04e3T^{2} \)
73 \( 1 + 42.2T + 5.32e3T^{2} \)
79 \( 1 - 32.2T + 6.24e3T^{2} \)
83 \( 1 - 19.0iT - 6.88e3T^{2} \)
89 \( 1 - 77.7iT - 7.92e3T^{2} \)
97 \( 1 - 123.T + 9.40e3T^{2} \)
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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

−12.42932915950689862805487322474, −11.19995500546271338464021117874, −10.24554779430369391398113281931, −8.818262972279586532660158387508, −7.967684207470289375916480237009, −6.27243028650125139217338778452, −6.19174551545272609254534515929, −4.19440087409026690479558000657, −2.26835838947699783922192194171, −0.74592230720582518964709234095, 2.86101600310500669062557593379, 3.66275522178063576004575429645, 5.89065697586729417822490326579, 6.39918656834608385935635783407, 7.34045804139117231839057179067, 9.089694051239487607643402733213, 10.17553173640992861856028974172, 10.95763381667937682856969214367, 11.39243391585098507272706853924, 12.78971170830545680770202130454

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