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Properties

Label 2-160-40.19-c4-0-9
Degree $2$
Conductor $160$
Sign $0.603 - 0.797i$
Analytic cond. $16.5391$
Root an. cond. $4.06684$
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  + 10.2i·3-s + (−17.9 − 17.4i)5-s + 84.8·7-s − 23.1·9-s − 71.3·11-s + 109.·13-s + (177. − 182. i)15-s − 151. i·17-s + 368.·19-s + 865. i·21-s + 358.·23-s + (16.8 + 624. i)25-s + 590. i·27-s + 387. i·29-s + 1.68e3i·31-s + ⋯
L(s)  = 1  + 1.13i·3-s + (−0.716 − 0.697i)5-s + 1.73·7-s − 0.285·9-s − 0.589·11-s + 0.646·13-s + (0.790 − 0.812i)15-s − 0.523i·17-s + 1.02·19-s + 1.96i·21-s + 0.676·23-s + (0.0269 + 0.999i)25-s + 0.810i·27-s + 0.460i·29-s + 1.75i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.603 - 0.797i$
Analytic conductor: \(16.5391\)
Root analytic conductor: \(4.06684\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :2),\ 0.603 - 0.797i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.80343 + 0.897000i\)
\(L(\frac12)\) \(\approx\) \(1.80343 + 0.897000i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (17.9 + 17.4i)T \)
good3 \( 1 - 10.2iT - 81T^{2} \)
7 \( 1 - 84.8T + 2.40e3T^{2} \)
11 \( 1 + 71.3T + 1.46e4T^{2} \)
13 \( 1 - 109.T + 2.85e4T^{2} \)
17 \( 1 + 151. iT - 8.35e4T^{2} \)
19 \( 1 - 368.T + 1.30e5T^{2} \)
23 \( 1 - 358.T + 2.79e5T^{2} \)
29 \( 1 - 387. iT - 7.07e5T^{2} \)
31 \( 1 - 1.68e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.15e3T + 1.87e6T^{2} \)
41 \( 1 + 2.54e3T + 2.82e6T^{2} \)
43 \( 1 - 1.35e3iT - 3.41e6T^{2} \)
47 \( 1 - 901.T + 4.87e6T^{2} \)
53 \( 1 - 3.40e3T + 7.89e6T^{2} \)
59 \( 1 - 1.45e3T + 1.21e7T^{2} \)
61 \( 1 + 5.32e3iT - 1.38e7T^{2} \)
67 \( 1 + 657. iT - 2.01e7T^{2} \)
71 \( 1 + 6.74e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.13e3iT - 2.83e7T^{2} \)
79 \( 1 + 2.30e3iT - 3.89e7T^{2} \)
83 \( 1 + 2.14e3iT - 4.74e7T^{2} \)
89 \( 1 - 5.11e3T + 6.27e7T^{2} \)
97 \( 1 - 9.16e3iT - 8.85e7T^{2} \)
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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

−12.06151281197882829495649520041, −11.23415252899902936913459246593, −10.52020821573441040155825766065, −9.167438183231429660007285542419, −8.353512170449444356540942243401, −7.39237361974203979368967402064, −5.08550044810980707015917115485, −4.86223929605148465555977339859, −3.48360990661710555179389504313, −1.24504847065828227518791032617, 1.00437875591018459466059527539, 2.35138360309546054606804760191, 4.14039761011167712352643206034, 5.62803349330889621039155606211, 7.06525281747640682501795036568, 7.78832032366844223920108451254, 8.431970469185465419649538882799, 10.32218211050857635435805804216, 11.41071113444472734184258341992, 11.78218690026861009636622556525

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