Properties

Label 2-4020-201.200-c1-0-50
Degree $2$
Conductor $4020$
Sign $0.857 + 0.515i$
Analytic cond. $32.0998$
Root an. cond. $5.66567$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.15 + 1.29i)3-s − 5-s − 2.00i·7-s + (−0.351 − 2.97i)9-s + 4.96·11-s − 1.09i·13-s + (1.15 − 1.29i)15-s + 2.56i·17-s − 2.48·19-s + (2.59 + 2.31i)21-s + 2.68i·23-s + 25-s + (4.26 + 2.97i)27-s + 0.905i·29-s − 0.150i·31-s + ⋯
L(s)  = 1  + (−0.664 + 0.747i)3-s − 0.447·5-s − 0.758i·7-s + (−0.117 − 0.993i)9-s + 1.49·11-s − 0.303i·13-s + (0.297 − 0.334i)15-s + 0.622i·17-s − 0.570·19-s + (0.567 + 0.504i)21-s + 0.560i·23-s + 0.200·25-s + (0.820 + 0.572i)27-s + 0.168i·29-s − 0.0270i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.857 + 0.515i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ 0.857 + 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.222475681\)
\(L(\frac12)\) \(\approx\) \(1.222475681\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.15 - 1.29i)T \)
5 \( 1 + T \)
67 \( 1 + (1.50 + 8.04i)T \)
good7 \( 1 + 2.00iT - 7T^{2} \)
11 \( 1 - 4.96T + 11T^{2} \)
13 \( 1 + 1.09iT - 13T^{2} \)
17 \( 1 - 2.56iT - 17T^{2} \)
19 \( 1 + 2.48T + 19T^{2} \)
23 \( 1 - 2.68iT - 23T^{2} \)
29 \( 1 - 0.905iT - 29T^{2} \)
31 \( 1 + 0.150iT - 31T^{2} \)
37 \( 1 + 8.64T + 37T^{2} \)
41 \( 1 - 9.39T + 41T^{2} \)
43 \( 1 - 1.68iT - 43T^{2} \)
47 \( 1 + 13.1iT - 47T^{2} \)
53 \( 1 - 1.64T + 53T^{2} \)
59 \( 1 - 7.45iT - 59T^{2} \)
61 \( 1 + 11.3iT - 61T^{2} \)
71 \( 1 + 0.805iT - 71T^{2} \)
73 \( 1 + 15.9T + 73T^{2} \)
79 \( 1 - 0.585iT - 79T^{2} \)
83 \( 1 + 8.06iT - 83T^{2} \)
89 \( 1 - 18.0iT - 89T^{2} \)
97 \( 1 + 6.66iT - 97T^{2} \)
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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

−8.591091168200577087979945610672, −7.51520649948079951809315643768, −6.82848717804113905965678321389, −6.19470435067541900585159145100, −5.37774006514614976629170231853, −4.39508932038266259934208890987, −3.91905796926497179926769625798, −3.30594146127388218905294003101, −1.62945072983580175477222657526, −0.51313907554585020323012671864, 0.898504420598864332490836833684, 1.93346512762620488781845595647, 2.87309999965461716545932666030, 4.09950138902247221954477124150, 4.73025214141822030944751567054, 5.78588797712520871127133959007, 6.26621488665940664910378743269, 7.04188522456357960905131349929, 7.56209806778095591401028916334, 8.715216583174692453750450036730

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