Properties

Label 2-4020-5.4-c1-0-32
Degree $2$
Conductor $4020$
Sign $0.769 - 0.639i$
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  + i·3-s + (−1.71 + 1.42i)5-s + 1.37i·7-s − 9-s + 3.21·11-s − 1.24i·13-s + (−1.42 − 1.71i)15-s − 5.00i·17-s − 0.139·19-s − 1.37·21-s + 1.01i·23-s + (0.915 − 4.91i)25-s i·27-s + 8.55·29-s + 1.37·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.769 + 0.639i)5-s + 0.521i·7-s − 0.333·9-s + 0.969·11-s − 0.345i·13-s + (−0.369 − 0.444i)15-s − 1.21i·17-s − 0.0319·19-s − 0.301·21-s + 0.210i·23-s + (0.183 − 0.983i)25-s − 0.192i·27-s + 1.58·29-s + 0.247·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.769 - 0.639i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (1609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ 0.769 - 0.639i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.664843900\)
\(L(\frac12)\) \(\approx\) \(1.664843900\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (1.71 - 1.42i)T \)
67 \( 1 + iT \)
good7 \( 1 - 1.37iT - 7T^{2} \)
11 \( 1 - 3.21T + 11T^{2} \)
13 \( 1 + 1.24iT - 13T^{2} \)
17 \( 1 + 5.00iT - 17T^{2} \)
19 \( 1 + 0.139T + 19T^{2} \)
23 \( 1 - 1.01iT - 23T^{2} \)
29 \( 1 - 8.55T + 29T^{2} \)
31 \( 1 - 1.37T + 31T^{2} \)
37 \( 1 - 5.04iT - 37T^{2} \)
41 \( 1 + 0.647T + 41T^{2} \)
43 \( 1 + 9.78iT - 43T^{2} \)
47 \( 1 + 3.18iT - 47T^{2} \)
53 \( 1 + 2.68iT - 53T^{2} \)
59 \( 1 - 9.87T + 59T^{2} \)
61 \( 1 - 12.9T + 61T^{2} \)
71 \( 1 + 15.7T + 71T^{2} \)
73 \( 1 + 16.0iT - 73T^{2} \)
79 \( 1 - 1.22T + 79T^{2} \)
83 \( 1 - 3.38iT - 83T^{2} \)
89 \( 1 - 2.10T + 89T^{2} \)
97 \( 1 + 8.14iT - 97T^{2} \)
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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

−8.641769315348333006214050038938, −7.84196787224610056269616437726, −6.96489676564764020279479565627, −6.46979266817399932954462721876, −5.45002874298777680090426251025, −4.70698581424501830458597093436, −3.87594584827128015822129630331, −3.14888375816440384478634702150, −2.36393143140803421151036162815, −0.71744161645113035907798860259, 0.818974267167133799677349584157, 1.56425801619460945369702120018, 2.86946278325175702629709034442, 4.05540643671525158437078316621, 4.25717104611919301476183479567, 5.42162702977408062504120041527, 6.34061974590015134083908449079, 6.90351476933184207458807469287, 7.65857811515848208780387675557, 8.434352193371690630344512488075

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