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Properties

Label 2-160-160.147-c1-0-20
Degree $2$
Conductor $160$
Sign $-0.938 + 0.344i$
Analytic cond. $1.27760$
Root an. cond. $1.13031$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.788 − 1.17i)2-s + (−1.91 + 0.793i)3-s + (−0.757 − 1.85i)4-s + (−2.19 − 0.424i)5-s + (−0.577 + 2.87i)6-s − 3.99i·7-s + (−2.77 − 0.569i)8-s + (0.916 − 0.916i)9-s + (−2.22 + 2.24i)10-s + (−3.60 + 1.49i)11-s + (2.91 + 2.94i)12-s + (4.61 − 1.91i)13-s + (−4.69 − 3.14i)14-s + (4.54 − 0.928i)15-s + (−2.85 + 2.80i)16-s + (2.42 − 2.42i)17-s + ⋯
L(s)  = 1  + (0.557 − 0.830i)2-s + (−1.10 + 0.457i)3-s + (−0.378 − 0.925i)4-s + (−0.981 − 0.189i)5-s + (−0.235 + 1.17i)6-s − 1.50i·7-s + (−0.979 − 0.201i)8-s + (0.305 − 0.305i)9-s + (−0.704 + 0.709i)10-s + (−1.08 + 0.450i)11-s + (0.842 + 0.849i)12-s + (1.27 − 0.529i)13-s + (−1.25 − 0.841i)14-s + (1.17 − 0.239i)15-s + (−0.713 + 0.701i)16-s + (0.587 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.938 + 0.344i$
Analytic conductor: \(1.27760\)
Root analytic conductor: \(1.13031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1/2),\ -0.938 + 0.344i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.109298 - 0.614792i\)
\(L(\frac12)\) \(\approx\) \(0.109298 - 0.614792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.788 + 1.17i)T \)
5 \( 1 + (2.19 + 0.424i)T \)
good3 \( 1 + (1.91 - 0.793i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + 3.99iT - 7T^{2} \)
11 \( 1 + (3.60 - 1.49i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-4.61 + 1.91i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + (-2.42 + 2.42i)T - 17iT^{2} \)
19 \( 1 + (2.25 + 0.934i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 - 1.48iT - 23T^{2} \)
29 \( 1 + (-6.67 - 2.76i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 4.27iT - 31T^{2} \)
37 \( 1 + (7.17 + 2.97i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (4.93 + 4.93i)T + 41iT^{2} \)
43 \( 1 + (-1.46 + 3.52i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (1.16 + 1.16i)T + 47iT^{2} \)
53 \( 1 + (4.27 + 1.77i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-4.20 + 1.74i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.256 + 0.618i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (0.333 + 0.804i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-9.01 - 9.01i)T + 71iT^{2} \)
73 \( 1 + 2.58T + 73T^{2} \)
79 \( 1 - 4.01iT - 79T^{2} \)
83 \( 1 + (-0.144 - 0.349i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (5.80 + 5.80i)T + 89iT^{2} \)
97 \( 1 + (-3.39 - 3.39i)T + 97iT^{2} \)
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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

−12.30409493008392636866278635977, −11.27905996305605410941114161286, −10.67722490000101759355247158062, −10.13298158266100970384446073188, −8.317631601105708720175571107107, −6.95976008964022372278328128175, −5.43868142175819040016198360238, −4.52972791695675518444810105845, −3.50331052138643609395757686036, −0.57830801419294047125899183973, 3.21073440669958717898858830459, 4.91009864948036148699320362667, 5.94909475880190377701835326948, 6.59679838208239660536674727609, 8.142343227792312799154573271417, 8.652488208709809405176996605883, 10.75253525558944727445258049924, 11.76793589408639925392266459757, 12.25037091637080363649206976423, 13.10366252896209115706690550800

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