Properties

Label 1-4020-4020.1499-r0-0-0
Degree $1$
Conductor $4020$
Sign $0.559 + 0.828i$
Analytic cond. $18.6688$
Root an. cond. $18.6688$
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  + (−0.959 + 0.281i)7-s + (0.415 − 0.909i)11-s + (0.654 + 0.755i)13-s + (0.841 + 0.540i)17-s + (0.959 + 0.281i)19-s + (0.142 + 0.989i)23-s − 29-s + (0.654 − 0.755i)31-s − 37-s + (−0.841 − 0.540i)41-s + (0.841 + 0.540i)43-s + (0.142 + 0.989i)47-s + (0.841 − 0.540i)49-s + (0.841 − 0.540i)53-s + (−0.654 + 0.755i)59-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)7-s + (0.415 − 0.909i)11-s + (0.654 + 0.755i)13-s + (0.841 + 0.540i)17-s + (0.959 + 0.281i)19-s + (0.142 + 0.989i)23-s − 29-s + (0.654 − 0.755i)31-s − 37-s + (−0.841 − 0.540i)41-s + (0.841 + 0.540i)43-s + (0.142 + 0.989i)47-s + (0.841 − 0.540i)49-s + (0.841 − 0.540i)53-s + (−0.654 + 0.755i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.559 + 0.828i$
Analytic conductor: \(18.6688\)
Root analytic conductor: \(18.6688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4020,\ (0:\ ),\ 0.559 + 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.387851800 + 0.7371293443i\)
\(L(\frac12)\) \(\approx\) \(1.387851800 + 0.7371293443i\)
\(L(1)\) \(\approx\) \(1.046274458 + 0.1283899169i\)
\(L(1)\) \(\approx\) \(1.046274458 + 0.1283899169i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
67 \( 1 \)
good7 \( 1 + (-0.959 + 0.281i)T \)
11 \( 1 + (0.415 - 0.909i)T \)
13 \( 1 + (0.654 + 0.755i)T \)
17 \( 1 + (0.841 + 0.540i)T \)
19 \( 1 + (0.959 + 0.281i)T \)
23 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 - T \)
31 \( 1 + (0.654 - 0.755i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.841 - 0.540i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (0.142 + 0.989i)T \)
53 \( 1 + (0.841 - 0.540i)T \)
59 \( 1 + (-0.654 + 0.755i)T \)
61 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (-0.415 - 0.909i)T \)
79 \( 1 + (0.654 + 0.755i)T \)
83 \( 1 + (-0.415 + 0.909i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.49615579498918929751159388270, −17.65154262162702915981557411068, −16.99460783735246497360415395391, −16.30932625895576583569753749577, −15.674336922670863643304016848086, −15.09838504731307855778154759240, −14.16176318496708245626437270299, −13.641747112662192780005021445408, −12.80623126374445376897256079948, −12.30820648914581664285861776341, −11.610496247349328185184779568495, −10.59140790296691832224147251748, −10.091043698500393896154318732, −9.43779472501723460955645106462, −8.727895598050960858959467821328, −7.79709413775917413910843397977, −7.059853417774514920600368480203, −6.56720830714521802523391642710, −5.5839306918278230963890058045, −4.981502079209416595614807056572, −3.90823292871422123000470653076, −3.328918430277604571282842966655, −2.57635906227646514914968348995, −1.42749147459843627540941288038, −0.5503928767408337577780777050, 0.927673377980015775269483103923, 1.7172565312832134448445845126, 2.89182828435596269965453938581, 3.56579227449297815034956775810, 4.02823785903330558267647596822, 5.42514130158056164531423111323, 5.83144736306714482383220275043, 6.54521640854017348649232785191, 7.35880737404305372584667016537, 8.15625530757431363559402436701, 9.03239294852916052429684922149, 9.448531440654940834459402664065, 10.2332060054955251851647893155, 11.08889865766770472456794228283, 11.777290001606603547572157928020, 12.301880464542288674663241042840, 13.30244029640732526132552536902, 13.692415449062070751736364444139, 14.4018407769425124201239575363, 15.33185073627072077743005374402, 15.90052237566664391260507121682, 16.60470425984265431213251952364, 16.964811044764535006726729371557, 18.0131095450369542110003810369, 18.77098519915282599086505544184

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