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Properties

Label 1-4024-4024.21-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.997 - 0.0665i$
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  + (−0.289 − 0.957i)3-s + (0.993 − 0.112i)5-s + (−0.965 + 0.259i)7-s + (−0.831 + 0.554i)9-s + (−0.0187 − 0.999i)11-s + (−0.168 + 0.985i)13-s + (−0.395 − 0.918i)15-s + (−0.787 + 0.615i)17-s + (0.00625 − 0.999i)19-s + (0.528 + 0.848i)21-s + (−0.883 − 0.468i)23-s + (0.974 − 0.223i)25-s + (0.772 + 0.635i)27-s + (0.977 − 0.211i)29-s + (0.939 − 0.343i)31-s + ⋯
L(s)  = 1  + (−0.289 − 0.957i)3-s + (0.993 − 0.112i)5-s + (−0.965 + 0.259i)7-s + (−0.831 + 0.554i)9-s + (−0.0187 − 0.999i)11-s + (−0.168 + 0.985i)13-s + (−0.395 − 0.918i)15-s + (−0.787 + 0.615i)17-s + (0.00625 − 0.999i)19-s + (0.528 + 0.848i)21-s + (−0.883 − 0.468i)23-s + (0.974 − 0.223i)25-s + (0.772 + 0.635i)27-s + (0.977 − 0.211i)29-s + (0.939 − 0.343i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.997 - 0.0665i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.997 - 0.0665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.215716260 - 0.04048147145i\)
\(L(\frac12)\) \(\approx\) \(1.215716260 - 0.04048147145i\)
\(L(1)\) \(\approx\) \(0.9045792299 - 0.2164844838i\)
\(L(1)\) \(\approx\) \(0.9045792299 - 0.2164844838i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.289 - 0.957i)T \)
5 \( 1 + (0.993 - 0.112i)T \)
7 \( 1 + (-0.965 + 0.259i)T \)
11 \( 1 + (-0.0187 - 0.999i)T \)
13 \( 1 + (-0.168 + 0.985i)T \)
17 \( 1 + (-0.787 + 0.615i)T \)
19 \( 1 + (0.00625 - 0.999i)T \)
23 \( 1 + (-0.883 - 0.468i)T \)
29 \( 1 + (0.977 - 0.211i)T \)
31 \( 1 + (0.939 - 0.343i)T \)
37 \( 1 + (-0.580 + 0.814i)T \)
41 \( 1 + (-0.229 + 0.973i)T \)
43 \( 1 + (-0.407 + 0.913i)T \)
47 \( 1 + (-0.845 - 0.533i)T \)
53 \( 1 + (0.00625 + 0.999i)T \)
59 \( 1 + (-0.360 + 0.932i)T \)
61 \( 1 + (0.951 + 0.307i)T \)
67 \( 1 + (0.395 - 0.918i)T \)
71 \( 1 + (0.713 + 0.700i)T \)
73 \( 1 + (-0.441 - 0.897i)T \)
79 \( 1 + (0.747 - 0.663i)T \)
83 \( 1 + (-0.810 + 0.585i)T \)
89 \( 1 + (-0.649 + 0.760i)T \)
97 \( 1 + (-0.325 + 0.945i)T \)
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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

−18.162326210647279179080492327975, −17.654147665377252487481521454609, −17.24132953654097949892699952559, −16.38593929271509160531961077777, −15.70219863787036286936608827861, −15.356843543284568936165106522730, −14.20901048879926022533019678790, −13.98536893652742823057719055390, −12.86801040137944970937821112482, −12.45232033682959721725949102603, −11.566323906062124030354811765950, −10.53375684048257323681291351705, −10.02652859134226637570000550284, −9.85481714913167446456623741348, −8.99079878077396488180141665714, −8.17073174495890563438139131451, −6.991822358537800409164618175160, −6.49708058413776771853836667879, −5.62336170599744752094968491627, −5.1355198035013052531884823604, −4.22825669843523248955339856155, −3.42175496214769496904507249304, −2.695741151879473378057953309127, −1.83371150066581585773914854037, −0.45542457077589888463214387404, 0.75688557621636255396528070176, 1.69608055711577601824722418254, 2.530990944476728847245909399834, 3.00367325447745690128267481253, 4.32789592471130127915016625450, 5.16104900975412809870881388270, 6.13590421068990280209135302004, 6.435390334971739401528292313175, 6.839398471671936178915277256876, 8.1871620354316028287376695214, 8.64504004143867185940979109982, 9.41774636256360525931339650932, 10.15504789663794118502325844630, 10.96651609838536501549165852564, 11.73059743875760214772249056260, 12.34123365134098219611626500111, 13.21108620110737365788445312020, 13.54890304579830068460302500042, 14.007326416842552281304786435472, 15.00478667137619455507345420010, 15.965692692026130340033711567181, 16.618884159246839857393154629621, 17.07535832631377507190610028506, 17.907529361130320857165782694146, 18.362682861576937922239283860179

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