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Properties

Label 2-4020-201.200-c1-0-17
Degree $2$
Conductor $4020$
Sign $0.359 - 0.933i$
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  + (0.296 − 1.70i)3-s − 5-s + 4.75i·7-s + (−2.82 − 1.01i)9-s + 4.74·11-s − 2.55i·13-s + (−0.296 + 1.70i)15-s + 2.48i·17-s + 1.87·19-s + (8.12 + 1.41i)21-s + 0.122i·23-s + 25-s + (−2.56 + 4.51i)27-s + 5.57i·29-s + 1.05i·31-s + ⋯
L(s)  = 1  + (0.171 − 0.985i)3-s − 0.447·5-s + 1.79i·7-s + (−0.941 − 0.337i)9-s + 1.43·11-s − 0.707i·13-s + (−0.0766 + 0.440i)15-s + 0.602i·17-s + 0.430·19-s + (1.77 + 0.308i)21-s + 0.0254i·23-s + 0.200·25-s + (−0.493 + 0.869i)27-s + 1.03i·29-s + 0.190i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.359 - 0.933i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.359 - 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.389558759\)
\(L(\frac12)\) \(\approx\) \(1.389558759\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.296 + 1.70i)T \)
5 \( 1 + T \)
67 \( 1 + (-8.02 - 1.59i)T \)
good7 \( 1 - 4.75iT - 7T^{2} \)
11 \( 1 - 4.74T + 11T^{2} \)
13 \( 1 + 2.55iT - 13T^{2} \)
17 \( 1 - 2.48iT - 17T^{2} \)
19 \( 1 - 1.87T + 19T^{2} \)
23 \( 1 - 0.122iT - 23T^{2} \)
29 \( 1 - 5.57iT - 29T^{2} \)
31 \( 1 - 1.05iT - 31T^{2} \)
37 \( 1 + 6.44T + 37T^{2} \)
41 \( 1 + 4.08T + 41T^{2} \)
43 \( 1 + 3.58iT - 43T^{2} \)
47 \( 1 + 4.59iT - 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 - 8.30iT - 59T^{2} \)
61 \( 1 - 0.253iT - 61T^{2} \)
71 \( 1 - 5.92iT - 71T^{2} \)
73 \( 1 - 9.97T + 73T^{2} \)
79 \( 1 - 11.8iT - 79T^{2} \)
83 \( 1 - 6.91iT - 83T^{2} \)
89 \( 1 - 13.5iT - 89T^{2} \)
97 \( 1 + 15.3iT - 97T^{2} \)
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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

−8.543145001878591873736496485244, −8.042101377376146715567959219921, −6.97782727636060568131821487692, −6.51046218182056547567592823399, −5.65513162834445496905517219334, −5.14888368752216688606127194532, −3.71441453797046327692481852277, −3.06842823228235807195284835927, −2.08701130904601680938930213116, −1.23022641267423940183113919463, 0.41354798039216010034280013365, 1.65164838986807661201757026579, 3.23095602302395373021076739229, 3.76627231569734005756663827476, 4.40727530586392730466523968132, 4.91723779962632217779267801654, 6.26844656951525260846802365422, 6.83162393427217391677724493750, 7.63025260622882748295907686174, 8.268396747799176760024092198526

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