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Properties

Label 1-4024-4024.1181-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.282 - 0.959i$
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  + (0.570 − 0.821i)3-s + (−0.452 − 0.891i)5-s + (0.889 + 0.457i)7-s + (−0.349 − 0.937i)9-s + (0.649 − 0.760i)11-s + (0.0812 + 0.996i)13-s + (−0.990 − 0.137i)15-s + (0.620 + 0.784i)17-s + (0.958 + 0.283i)19-s + (0.883 − 0.468i)21-s + (−0.894 − 0.446i)23-s + (−0.590 + 0.806i)25-s + (−0.968 − 0.247i)27-s + (0.934 − 0.355i)29-s + (−0.915 + 0.401i)31-s + ⋯
L(s)  = 1  + (0.570 − 0.821i)3-s + (−0.452 − 0.891i)5-s + (0.889 + 0.457i)7-s + (−0.349 − 0.937i)9-s + (0.649 − 0.760i)11-s + (0.0812 + 0.996i)13-s + (−0.990 − 0.137i)15-s + (0.620 + 0.784i)17-s + (0.958 + 0.283i)19-s + (0.883 − 0.468i)21-s + (−0.894 − 0.446i)23-s + (−0.590 + 0.806i)25-s + (−0.968 − 0.247i)27-s + (0.934 − 0.355i)29-s + (−0.915 + 0.401i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.282 - 0.959i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.282 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.984891557 - 1.485425745i\)
\(L(\frac12)\) \(\approx\) \(1.984891557 - 1.485425745i\)
\(L(1)\) \(\approx\) \(1.322878512 - 0.5355948209i\)
\(L(1)\) \(\approx\) \(1.322878512 - 0.5355948209i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.570 - 0.821i)T \)
5 \( 1 + (-0.452 - 0.891i)T \)
7 \( 1 + (0.889 + 0.457i)T \)
11 \( 1 + (0.649 - 0.760i)T \)
13 \( 1 + (0.0812 + 0.996i)T \)
17 \( 1 + (0.620 + 0.784i)T \)
19 \( 1 + (0.958 + 0.283i)T \)
23 \( 1 + (-0.894 - 0.446i)T \)
29 \( 1 + (0.934 - 0.355i)T \)
31 \( 1 + (-0.915 + 0.401i)T \)
37 \( 1 + (-0.974 - 0.223i)T \)
41 \( 1 + (0.337 + 0.941i)T \)
43 \( 1 + (0.905 + 0.424i)T \)
47 \( 1 + (0.713 + 0.700i)T \)
53 \( 1 + (0.958 - 0.283i)T \)
59 \( 1 + (-0.289 - 0.957i)T \)
61 \( 1 + (0.253 - 0.967i)T \)
67 \( 1 + (0.990 - 0.137i)T \)
71 \( 1 + (-0.418 - 0.908i)T \)
73 \( 1 + (0.560 + 0.828i)T \)
79 \( 1 + (-0.395 - 0.918i)T \)
83 \( 1 + (0.871 - 0.490i)T \)
89 \( 1 + (-0.441 - 0.897i)T \)
97 \( 1 + (0.900 - 0.435i)T \)
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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

−18.52852849029197109531487433044, −17.912756265042983004849916509608, −17.36128632687829723868294311792, −16.43448074351077934849429521222, −15.653544280908990192923021555698, −15.26588942856377308370650626379, −14.4640989134740950904899447680, −14.10256113048026543816184586186, −13.516886206335185450045412918164, −12.135094886931380107077652430045, −11.75960131266302251540991662123, −10.81889025081202847906788162803, −10.384338050033878787590407359258, −9.7495783917200348787967513982, −8.91816510615036967905288506681, −8.07084060939136440766749358741, −7.41577231772155968538507539913, −7.08072759353387534113594095372, −5.63032492355903003239010796816, −5.14154068945434450073075090990, −4.07918420464825209395424698015, −3.73942645951534991245237566388, −2.829377247177100301941014232655, −2.113166860037777479364854362626, −0.92278685149858099959139128294, 0.823281267822629757198859091450, 1.50451146421381442527362672175, 2.11023301668708648923434400724, 3.32696932049291249867242341227, 3.94164730264724047928372563281, 4.783729708492956015323961416178, 5.75465406077018860859034061808, 6.28714584267385446448383833401, 7.365055294082289442085205898143, 7.96800203669890376672806560895, 8.5576496507792187893567862954, 9.00377644314274742229917246211, 9.77837019685926463874572550019, 11.061617452063830862096609946148, 11.687344605680008775711035824451, 12.20552007486090389351116464851, 12.66883876700857105951393509933, 13.75229276045259080344971305316, 14.2148826090697315655287914237, 14.63395698802487284530370759982, 15.72006190949768592432821561827, 16.24853271080432803619207858703, 17.08266623896878080416185766536, 17.64896335137737368054462220366, 18.52868287158774943815916500961

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