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Properties

Label 1-4024-4024.1283-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.921 + 0.388i$
Analytic cond. $18.6873$
Root an. cond. $18.6873$
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  + (0.677 − 0.735i)3-s + (0.659 − 0.752i)5-s + (0.590 − 0.806i)7-s + (−0.0812 − 0.996i)9-s + (−0.372 + 0.928i)11-s + (−0.289 − 0.957i)13-s + (−0.106 − 0.994i)15-s + (−0.992 + 0.124i)17-s + (−0.795 + 0.605i)19-s + (−0.192 − 0.981i)21-s + (−0.877 + 0.479i)23-s + (−0.131 − 0.991i)25-s + (−0.787 − 0.615i)27-s + (−0.990 − 0.137i)29-s + (0.313 − 0.949i)31-s + ⋯
L(s)  = 1  + (0.677 − 0.735i)3-s + (0.659 − 0.752i)5-s + (0.590 − 0.806i)7-s + (−0.0812 − 0.996i)9-s + (−0.372 + 0.928i)11-s + (−0.289 − 0.957i)13-s + (−0.106 − 0.994i)15-s + (−0.992 + 0.124i)17-s + (−0.795 + 0.605i)19-s + (−0.192 − 0.981i)21-s + (−0.877 + 0.479i)23-s + (−0.131 − 0.991i)25-s + (−0.787 − 0.615i)27-s + (−0.990 − 0.137i)29-s + (0.313 − 0.949i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.921 + 0.388i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.921 + 0.388i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2727947542 - 1.350751475i\)
\(L(\frac12)\) \(\approx\) \(-0.2727947542 - 1.350751475i\)
\(L(1)\) \(\approx\) \(1.020540375 - 0.7001789044i\)
\(L(1)\) \(\approx\) \(1.020540375 - 0.7001789044i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.677 - 0.735i)T \)
5 \( 1 + (0.659 - 0.752i)T \)
7 \( 1 + (0.590 - 0.806i)T \)
11 \( 1 + (-0.372 + 0.928i)T \)
13 \( 1 + (-0.289 - 0.957i)T \)
17 \( 1 + (-0.992 + 0.124i)T \)
19 \( 1 + (-0.795 + 0.605i)T \)
23 \( 1 + (-0.877 + 0.479i)T \)
29 \( 1 + (-0.990 - 0.137i)T \)
31 \( 1 + (0.313 - 0.949i)T \)
37 \( 1 + (-0.0312 - 0.999i)T \)
41 \( 1 + (-0.496 - 0.868i)T \)
43 \( 1 + (0.930 + 0.366i)T \)
47 \( 1 + (0.441 + 0.897i)T \)
53 \( 1 + (0.795 + 0.605i)T \)
59 \( 1 + (0.764 - 0.644i)T \)
61 \( 1 + (-0.429 + 0.902i)T \)
67 \( 1 + (-0.106 + 0.994i)T \)
71 \( 1 + (0.560 + 0.828i)T \)
73 \( 1 + (-0.925 - 0.378i)T \)
79 \( 1 + (-0.994 - 0.0999i)T \)
83 \( 1 + (-0.630 + 0.776i)T \)
89 \( 1 + (0.277 - 0.960i)T \)
97 \( 1 + (-0.998 - 0.0625i)T \)
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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

−18.7679392122837077725091216981, −18.367703500817661449507148399056, −17.50825865904113206705736975926, −16.75717384208922201984644670944, −16.03859942166146896212459741989, −15.25479322267462488648078522342, −14.87035506734235520925199623685, −14.068841248823817102353800534677, −13.69815398684291175235658395295, −12.92626989237893223064328218509, −11.73360150225204611847510618870, −11.1799373199584531374078285166, −10.5704892964612415654069953490, −9.8826365151549960782474505457, −8.910582516180015577588598230657, −8.773235204206327990802081693914, −7.86106094826656276233159242204, −6.86460869851337224126062511860, −6.18496044889991832375093623715, −5.321628782995314090707872931521, −4.66921347643434525843175908012, −3.82122833341763519580501744399, −2.86073917383008262888724431651, −2.312096392320691274064840941764, −1.75398110471919769991863083873, 0.29465512541600618564698102553, 1.36586754467353221088323490814, 2.04990906672730424515022248028, 2.56299374368489693716241170553, 4.04067093339082774507046455711, 4.27085611461877826514701262709, 5.49314065561093958352954031482, 6.007950882225892004042322736655, 7.12812953241928472956742182370, 7.5993134959588375607517568763, 8.2663461853015911514141580134, 8.93729038872683928155523493465, 9.799062274172355606713500674036, 10.295346721693016299796486832317, 11.22521969886431123152097970165, 12.21709970153056699953663624839, 12.80665371577403348478627596331, 13.22582998652869102424667923762, 13.89843235549428190819821503808, 14.60155487656883304118752375252, 15.21119834523963275695658778662, 15.99069519097316900118110015061, 17.125009229613405369498132245956, 17.47353426916692741841384162455, 17.890367626327194234021506073738

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