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Properties

Label 2-160-5.4-c3-0-2
Degree $2$
Conductor $160$
Sign $-0.963 - 0.268i$
Analytic cond. $9.44030$
Root an. cond. $3.07250$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·3-s + (−3 + 10.7i)5-s + 4i·7-s + 11·9-s − 43.0·11-s + 21.5i·13-s + (−43.0 − 12i)15-s − 43.0i·17-s − 129.·19-s − 16·21-s + 52i·23-s + (−106. − 64.6i)25-s + 152i·27-s + 158·29-s − 172.·31-s + ⋯
L(s)  = 1  + 0.769i·3-s + (−0.268 + 0.963i)5-s + 0.215i·7-s + 0.407·9-s − 1.18·11-s + 0.459i·13-s + (−0.741 − 0.206i)15-s − 0.614i·17-s − 1.56·19-s − 0.166·21-s + 0.471i·23-s + (−0.855 − 0.516i)25-s + 1.08i·27-s + 1.01·29-s − 0.998·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.268i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.963 - 0.268i$
Analytic conductor: \(9.44030\)
Root analytic conductor: \(3.07250\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :3/2),\ -0.963 - 0.268i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.135829 + 0.993849i\)
\(L(\frac12)\) \(\approx\) \(0.135829 + 0.993849i\)
\(L(\frac{5}{2})\) 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 \( 1 + (3 - 10.7i)T \)
good3 \( 1 - 4iT - 27T^{2} \)
7 \( 1 - 4iT - 343T^{2} \)
11 \( 1 + 43.0T + 1.33e3T^{2} \)
13 \( 1 - 21.5iT - 2.19e3T^{2} \)
17 \( 1 + 43.0iT - 4.91e3T^{2} \)
19 \( 1 + 129.T + 6.85e3T^{2} \)
23 \( 1 - 52iT - 1.21e4T^{2} \)
29 \( 1 - 158T + 2.43e4T^{2} \)
31 \( 1 + 172.T + 2.97e4T^{2} \)
37 \( 1 - 280. iT - 5.06e4T^{2} \)
41 \( 1 + 170T + 6.89e4T^{2} \)
43 \( 1 + 316iT - 7.95e4T^{2} \)
47 \( 1 - 244iT - 1.03e5T^{2} \)
53 \( 1 - 495. iT - 1.48e5T^{2} \)
59 \( 1 - 646.T + 2.05e5T^{2} \)
61 \( 1 - 82T + 2.26e5T^{2} \)
67 \( 1 - 692iT - 3.00e5T^{2} \)
71 \( 1 - 947.T + 3.57e5T^{2} \)
73 \( 1 + 430. iT - 3.89e5T^{2} \)
79 \( 1 - 344.T + 4.93e5T^{2} \)
83 \( 1 + 940iT - 5.71e5T^{2} \)
89 \( 1 + 6T + 7.04e5T^{2} \)
97 \( 1 - 1.07e3iT - 9.12e5T^{2} \)
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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

−12.89584786717290638240301133774, −11.69398976543697731619064335199, −10.61989375773241053007623833380, −10.16715855602592481812293053158, −8.887327729557964523310200075615, −7.61275506297247953397523927158, −6.56257294597258001810564248767, −5.07792270804021463522386212568, −3.86938786102425746351579754493, −2.47295271190011491138840026434, 0.45488452933802413072555484572, 2.07465652426443788323769838947, 4.10249846907163505199746653700, 5.35018745590059870492832433022, 6.71655501786741671270772032708, 7.917098763018298539358160331956, 8.520571410223212433306103359710, 10.02532160948514284831773894093, 10.95695219732599547207072155187, 12.48290605513198360932395087499

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