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Properties

Label 2-320-40.3-c1-0-10
Degree $2$
Conductor $320$
Sign $-0.973 - 0.229i$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)3-s + (−1 + 2i)5-s + (−1 − i)7-s − i·9-s − 4·11-s + (−3 + 3i)13-s + (3 − i)15-s + (−3 + 3i)17-s − 6i·19-s + 2i·21-s + (−3 + 3i)23-s + (−3 − 4i)25-s + (−4 + 4i)27-s + 2·29-s + 6i·31-s + ⋯
L(s)  = 1  + (−0.577 − 0.577i)3-s + (−0.447 + 0.894i)5-s + (−0.377 − 0.377i)7-s − 0.333i·9-s − 1.20·11-s + (−0.832 + 0.832i)13-s + (0.774 − 0.258i)15-s + (−0.727 + 0.727i)17-s − 1.37i·19-s + 0.436i·21-s + (−0.625 + 0.625i)23-s + (−0.600 − 0.800i)25-s + (−0.769 + 0.769i)27-s + 0.371·29-s + 1.07i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.973 - 0.229i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1 - 2i)T \)
good3 \( 1 + (1 + i)T + 3iT^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 + (3 - 3i)T - 17iT^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + (3 - 3i)T - 23iT^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 6iT - 31T^{2} \)
37 \( 1 + (3 + 3i)T + 37iT^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + (-3 - 3i)T + 43iT^{2} \)
47 \( 1 + (9 + 9i)T + 47iT^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 - 10iT - 59T^{2} \)
61 \( 1 + 12iT - 61T^{2} \)
67 \( 1 + (-9 + 9i)T - 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (-5 - 5i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-3 - 3i)T + 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (7 - 7i)T - 97iT^{2} \)
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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

−11.18185980305712437639796722003, −10.42895204673257971173588717182, −9.387384252825253987340778726341, −8.029120002825090631820258281723, −6.93738955347480394716968290260, −6.64235707921365981905084689582, −5.19357968208382330038713759751, −3.79198998007098028885591972392, −2.39267600089662764382802844966, 0, 2.57119187761471274617756645711, 4.28202416829487687819221214242, 5.14948456479960198295978311814, 5.90140119677323670090460866986, 7.63947010131915700151856943361, 8.228006242855630454386827497975, 9.549039434478250297792296135029, 10.24952265206505591234630510982, 11.16334154463584819392357277984

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