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Properties

Label 2-160-5.2-c2-0-5
Degree $2$
Conductor $160$
Sign $0.999 - 0.0219i$
Analytic cond. $4.35968$
Root an. cond. $2.08798$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.29 − 2.29i)3-s + (1.25 + 4.83i)5-s + (4.80 + 4.80i)7-s − 1.51i·9-s + 10.7·11-s + (1.07 − 1.07i)13-s + (13.9 + 8.21i)15-s + (−21.8 − 21.8i)17-s − 12.1i·19-s + 22.0·21-s + (21.9 − 21.9i)23-s + (−21.8 + 12.1i)25-s + (17.1 + 17.1i)27-s + 29.5i·29-s − 50.7·31-s + ⋯
L(s)  = 1  + (0.764 − 0.764i)3-s + (0.251 + 0.967i)5-s + (0.686 + 0.686i)7-s − 0.167i·9-s + 0.979·11-s + (0.0826 − 0.0826i)13-s + (0.931 + 0.547i)15-s + (−1.28 − 1.28i)17-s − 0.639i·19-s + 1.04·21-s + (0.955 − 0.955i)23-s + (−0.873 + 0.485i)25-s + (0.635 + 0.635i)27-s + 1.01i·29-s − 1.63·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0219i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.999 - 0.0219i$
Analytic conductor: \(4.35968\)
Root analytic conductor: \(2.08798\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1),\ 0.999 - 0.0219i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.03692 + 0.0223165i\)
\(L(\frac12)\) \(\approx\) \(2.03692 + 0.0223165i\)
\(L(2)\) 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 \( 1 + (-1.25 - 4.83i)T \)
good3 \( 1 + (-2.29 + 2.29i)T - 9iT^{2} \)
7 \( 1 + (-4.80 - 4.80i)T + 49iT^{2} \)
11 \( 1 - 10.7T + 121T^{2} \)
13 \( 1 + (-1.07 + 1.07i)T - 169iT^{2} \)
17 \( 1 + (21.8 + 21.8i)T + 289iT^{2} \)
19 \( 1 + 12.1iT - 361T^{2} \)
23 \( 1 + (-21.9 + 21.9i)T - 529iT^{2} \)
29 \( 1 - 29.5iT - 841T^{2} \)
31 \( 1 + 50.7T + 961T^{2} \)
37 \( 1 + (-23.6 - 23.6i)T + 1.36e3iT^{2} \)
41 \( 1 + 1.86T + 1.68e3T^{2} \)
43 \( 1 + (7.40 - 7.40i)T - 1.84e3iT^{2} \)
47 \( 1 + (32.1 + 32.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (4.22 - 4.22i)T - 2.80e3iT^{2} \)
59 \( 1 + 110. iT - 3.48e3T^{2} \)
61 \( 1 + 76.2T + 3.72e3T^{2} \)
67 \( 1 + (-31.6 - 31.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 99.3T + 5.04e3T^{2} \)
73 \( 1 + (45.9 - 45.9i)T - 5.32e3iT^{2} \)
79 \( 1 - 43.0iT - 6.24e3T^{2} \)
83 \( 1 + (-92.2 + 92.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 75.0iT - 7.92e3T^{2} \)
97 \( 1 + (-61.7 - 61.7i)T + 9.40e3iT^{2} \)
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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

−12.85871086648325716807613314488, −11.53312450565260347587793947009, −10.92018498694184911068490467553, −9.283529034687329531019965516368, −8.599461549939938779308209246460, −7.25478061015246314943323230677, −6.62079486411811878214807065400, −4.94542602736034555040171733211, −3.00306462783501739654298066324, −1.93953871168627208330473652232, 1.58599562100525790359544753947, 3.82692756942136162785624770382, 4.49766404717025990719368352674, 6.08029436790642699763503906438, 7.69442612722447085626781393439, 8.839849353911305508484298971718, 9.304594875962253196893193922927, 10.52592595056621537240873440918, 11.57055837167245238979610470300, 12.82449510689377895888613322548

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