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Properties

Label 2-3920-980.519-c0-0-1
Degree $2$
Conductor $3920$
Sign $-0.871 + 0.490i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.433 − 1.90i)3-s + (0.222 + 0.974i)5-s + (0.781 − 0.623i)7-s + (−2.52 + 1.21i)9-s + (1.75 − 0.846i)15-s + (−1.52 − 1.21i)21-s + (−1.21 − 1.52i)23-s + (−0.900 + 0.433i)25-s + (2.19 + 2.74i)27-s + (0.777 − 0.974i)29-s + (0.781 + 0.623i)35-s + (−0.400 − 1.75i)41-s + (0.193 − 0.846i)43-s + (−1.74 − 2.19i)45-s + (−1.75 − 0.846i)47-s + ⋯
L(s)  = 1  + (−0.433 − 1.90i)3-s + (0.222 + 0.974i)5-s + (0.781 − 0.623i)7-s + (−2.52 + 1.21i)9-s + (1.75 − 0.846i)15-s + (−1.52 − 1.21i)21-s + (−1.21 − 1.52i)23-s + (−0.900 + 0.433i)25-s + (2.19 + 2.74i)27-s + (0.777 − 0.974i)29-s + (0.781 + 0.623i)35-s + (−0.400 − 1.75i)41-s + (0.193 − 0.846i)43-s + (−1.74 − 2.19i)45-s + (−1.75 − 0.846i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-0.871 + 0.490i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (2479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ -0.871 + 0.490i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9518208629\)
\(L(\frac12)\) \(\approx\) \(0.9518208629\)
\(L(1)\) 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 \( 1 + (-0.222 - 0.974i)T \)
7 \( 1 + (-0.781 + 0.623i)T \)
good3 \( 1 + (0.433 + 1.90i)T + (-0.900 + 0.433i)T^{2} \)
11 \( 1 + (-0.623 - 0.781i)T^{2} \)
13 \( 1 + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (0.222 + 0.974i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (1.21 + 1.52i)T + (-0.222 + 0.974i)T^{2} \)
29 \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.222 + 0.974i)T^{2} \)
41 \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \)
43 \( 1 + (-0.193 + 0.846i)T + (-0.900 - 0.433i)T^{2} \)
47 \( 1 + (1.75 + 0.846i)T + (0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.222 - 0.974i)T^{2} \)
59 \( 1 + (0.900 + 0.433i)T^{2} \)
61 \( 1 + (-0.277 + 0.347i)T + (-0.222 - 0.974i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.623 + 0.781i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-1.40 + 0.678i)T + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.182029069633625406756150925212, −7.42870050454100778211000839817, −6.92069140761035104750066702667, −6.32340870942257183580325787031, −5.70686668149836832176437208069, −4.71682723768485562391969072537, −3.49532987844033660693507860589, −2.28418797343952218187227656148, −1.94034876758267210907675017420, −0.56026929182649568468690408122, 1.50240872484030482504091422657, 2.90173976585988230639534834357, 3.80085033021218206614568683505, 4.69272591980936374200739468752, 4.96891028025994132551851277111, 5.72801660239782599360397062101, 6.28952557321556496398115935224, 7.928754486611322472609497400042, 8.387059753036588457746030094404, 9.125812896460376698884526672810

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