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Properties

Label 2-160-20.7-c3-0-6
Degree $2$
Conductor $160$
Sign $-0.199 - 0.979i$
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  + (6.13 + 6.13i)3-s + (3.85 − 10.4i)5-s + (−24.7 + 24.7i)7-s + 48.1i·9-s + 39.8i·11-s + (3.91 − 3.91i)13-s + (87.9 − 40.6i)15-s + (42.3 + 42.3i)17-s + 61.0·19-s − 303.·21-s + (41.1 + 41.1i)23-s + (−95.2 − 80.9i)25-s + (−129. + 129. i)27-s − 57.2i·29-s − 228. i·31-s + ⋯
L(s)  = 1  + (1.17 + 1.17i)3-s + (0.345 − 0.938i)5-s + (−1.33 + 1.33i)7-s + 1.78i·9-s + 1.09i·11-s + (0.0835 − 0.0835i)13-s + (1.51 − 0.699i)15-s + (0.603 + 0.603i)17-s + 0.736·19-s − 3.15·21-s + (0.372 + 0.372i)23-s + (−0.761 − 0.647i)25-s + (−0.924 + 0.924i)27-s − 0.366i·29-s − 1.32i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.199 - 0.979i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.199 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.199 - 0.979i$
Analytic conductor: \(9.44030\)
Root analytic conductor: \(3.07250\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :3/2),\ -0.199 - 0.979i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.38444 + 1.69520i\)
\(L(\frac12)\) \(\approx\) \(1.38444 + 1.69520i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.85 + 10.4i)T \)
good3 \( 1 + (-6.13 - 6.13i)T + 27iT^{2} \)
7 \( 1 + (24.7 - 24.7i)T - 343iT^{2} \)
11 \( 1 - 39.8iT - 1.33e3T^{2} \)
13 \( 1 + (-3.91 + 3.91i)T - 2.19e3iT^{2} \)
17 \( 1 + (-42.3 - 42.3i)T + 4.91e3iT^{2} \)
19 \( 1 - 61.0T + 6.85e3T^{2} \)
23 \( 1 + (-41.1 - 41.1i)T + 1.21e4iT^{2} \)
29 \( 1 + 57.2iT - 2.43e4T^{2} \)
31 \( 1 + 228. iT - 2.97e4T^{2} \)
37 \( 1 + (-231. - 231. i)T + 5.06e4iT^{2} \)
41 \( 1 - 78.1T + 6.89e4T^{2} \)
43 \( 1 + (147. + 147. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-34.7 + 34.7i)T - 1.03e5iT^{2} \)
53 \( 1 + (-132. + 132. i)T - 1.48e5iT^{2} \)
59 \( 1 + 275.T + 2.05e5T^{2} \)
61 \( 1 + 322.T + 2.26e5T^{2} \)
67 \( 1 + (-89.6 + 89.6i)T - 3.00e5iT^{2} \)
71 \( 1 + 292. iT - 3.57e5T^{2} \)
73 \( 1 + (-299. + 299. i)T - 3.89e5iT^{2} \)
79 \( 1 - 227.T + 4.93e5T^{2} \)
83 \( 1 + (-287. - 287. i)T + 5.71e5iT^{2} \)
89 \( 1 + 471. iT - 7.04e5T^{2} \)
97 \( 1 + (-1.20e3 - 1.20e3i)T + 9.12e5iT^{2} \)
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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

−12.83326400786682281433026953230, −11.92808671908459148613566774020, −10.01895452225938394840557352162, −9.587963114167670564685050270951, −8.971639218404035033984019509144, −7.907814093646970764767799608100, −5.99509718575553152122499575192, −4.85076643559975961732569745604, −3.52498072883615748661478342057, −2.30889222880753008753700742543, 0.936775969244094641675154143975, 2.91132666807452457208722421842, 3.46337636991114703350129625498, 6.16360751913081504443246705156, 7.02511880364634428073786365619, 7.63807833144255604186425339161, 9.038899959634906932318867644309, 9.992410212212809358148553952183, 11.06063576893591994897464999263, 12.54218373782754652652701246212

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