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Properties

Label 2-160-40.19-c4-0-1
Degree $2$
Conductor $160$
Sign $-0.748 + 0.663i$
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  + 16.4i·3-s + (15.0 − 19.9i)5-s + 3.44·7-s − 188.·9-s − 123.·11-s − 198.·13-s + (327. + 247. i)15-s + 203. i·17-s − 78.3·19-s + 56.5i·21-s − 644.·23-s + (−171. − 600. i)25-s − 1.76e3i·27-s − 855. i·29-s + 1.08e3i·31-s + ⋯
L(s)  = 1  + 1.82i·3-s + (0.602 − 0.798i)5-s + 0.0702·7-s − 2.32·9-s − 1.01·11-s − 1.17·13-s + (1.45 + 1.09i)15-s + 0.705i·17-s − 0.217·19-s + 0.128i·21-s − 1.21·23-s + (−0.274 − 0.961i)25-s − 2.41i·27-s − 1.01i·29-s + 1.12i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 + 0.663i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.748 + 0.663i$
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: \((2,\ 160,\ (\ :2),\ -0.748 + 0.663i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.148677 - 0.391965i\)
\(L(\frac12)\) \(\approx\) \(0.148677 - 0.391965i\)
\(L(3)\) 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 \( 1 + (-15.0 + 19.9i)T \)
good3 \( 1 - 16.4iT - 81T^{2} \)
7 \( 1 - 3.44T + 2.40e3T^{2} \)
11 \( 1 + 123.T + 1.46e4T^{2} \)
13 \( 1 + 198.T + 2.85e4T^{2} \)
17 \( 1 - 203. iT - 8.35e4T^{2} \)
19 \( 1 + 78.3T + 1.30e5T^{2} \)
23 \( 1 + 644.T + 2.79e5T^{2} \)
29 \( 1 + 855. iT - 7.07e5T^{2} \)
31 \( 1 - 1.08e3iT - 9.23e5T^{2} \)
37 \( 1 - 546.T + 1.87e6T^{2} \)
41 \( 1 - 370.T + 2.82e6T^{2} \)
43 \( 1 - 257. iT - 3.41e6T^{2} \)
47 \( 1 - 315.T + 4.87e6T^{2} \)
53 \( 1 - 4.12e3T + 7.89e6T^{2} \)
59 \( 1 + 3.03e3T + 1.21e7T^{2} \)
61 \( 1 - 3.00e3iT - 1.38e7T^{2} \)
67 \( 1 - 4.35e3iT - 2.01e7T^{2} \)
71 \( 1 - 3.88e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.30e3iT - 2.83e7T^{2} \)
79 \( 1 + 6.52e3iT - 3.89e7T^{2} \)
83 \( 1 - 411. iT - 4.74e7T^{2} \)
89 \( 1 - 614.T + 6.27e7T^{2} \)
97 \( 1 + 4.93e3iT - 8.85e7T^{2} \)
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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

−12.77743843834288966534233917442, −11.67113040327511098016453200237, −10.28950465310804489923534893468, −10.04777277771136865430818065485, −8.966694539855044294064815184774, −8.022897410286907104661336472989, −5.87607624046087597255191834779, −5.02253552502989627856005056679, −4.15121881477229878810457124849, −2.50773734548581531544003419413, 0.14444346072548475134536317362, 1.99028566044398440420984409903, 2.78266435617565561116103369539, 5.35392964742517081196894666425, 6.39913215667656453673100817289, 7.35021502031350879691035009751, 7.978987457947813769420734765657, 9.529244877032039480595744720339, 10.74597032803858248226389925142, 11.82338512389172643872186797222

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