Properties

Label 2-160-20.19-c2-0-8
Degree $2$
Conductor $160$
Sign $-0.248 + 0.968i$
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  − 2.75·3-s + (4.30 − 2.54i)5-s − 3.84·7-s − 1.41·9-s − 6.19i·11-s − 16.1i·13-s + (−11.8 + 7.01i)15-s − 5.20i·17-s − 36.2i·19-s + 10.6·21-s − 22.0·23-s + (12.0 − 21.9i)25-s + 28.6·27-s − 20.0·29-s + 26.4i·31-s + ⋯
L(s)  = 1  − 0.918·3-s + (0.860 − 0.509i)5-s − 0.549·7-s − 0.157·9-s − 0.562i·11-s − 1.23i·13-s + (−0.789 + 0.467i)15-s − 0.306i·17-s − 1.90i·19-s + 0.504·21-s − 0.958·23-s + (0.480 − 0.876i)25-s + 1.06·27-s − 0.690·29-s + 0.852i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.248 + 0.968i$
Analytic conductor: \(4.35968\)
Root analytic conductor: \(2.08798\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1),\ -0.248 + 0.968i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.525821 - 0.677466i\)
\(L(\frac12)\) \(\approx\) \(0.525821 - 0.677466i\)
\(L(2)\) 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 \( 1 + (-4.30 + 2.54i)T \)
good3 \( 1 + 2.75T + 9T^{2} \)
7 \( 1 + 3.84T + 49T^{2} \)
11 \( 1 + 6.19iT - 121T^{2} \)
13 \( 1 + 16.1iT - 169T^{2} \)
17 \( 1 + 5.20iT - 289T^{2} \)
19 \( 1 + 36.2iT - 361T^{2} \)
23 \( 1 + 22.0T + 529T^{2} \)
29 \( 1 + 20.0T + 841T^{2} \)
31 \( 1 - 26.4iT - 961T^{2} \)
37 \( 1 - 69.3iT - 1.36e3T^{2} \)
41 \( 1 - 11.6T + 1.68e3T^{2} \)
43 \( 1 - 25.8T + 1.84e3T^{2} \)
47 \( 1 - 66.1T + 2.20e3T^{2} \)
53 \( 1 + 39.5iT - 2.80e3T^{2} \)
59 \( 1 + 27.7iT - 3.48e3T^{2} \)
61 \( 1 + 54.1T + 3.72e3T^{2} \)
67 \( 1 - 107.T + 4.48e3T^{2} \)
71 \( 1 + 70.7iT - 5.04e3T^{2} \)
73 \( 1 - 37.4iT - 5.32e3T^{2} \)
79 \( 1 - 97.6iT - 6.24e3T^{2} \)
83 \( 1 + 126.T + 6.88e3T^{2} \)
89 \( 1 - 133.T + 7.92e3T^{2} \)
97 \( 1 - 6.40iT - 9.40e3T^{2} \)
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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

−12.42791401961808559293218761106, −11.32090755192456688871612341165, −10.43642510652884672516532383341, −9.411382394815086863923385470770, −8.345284036083187973425926691601, −6.69819707755068470870634461382, −5.77112100372828806697192659757, −4.95242497740967389422903031408, −2.88370511693990816197480786143, −0.58515994233888709157297048897, 2.04581936630511967509081581448, 3.99706004168435385197973843922, 5.73407605288957058125913805273, 6.21887588966871315317851978939, 7.46007082544181204321768855922, 9.149270778872550061275279066661, 10.03340465985940141640169968100, 10.89637454016258455227434985113, 11.96541159501004482699646975682, 12.74300529885525526742693652829

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