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Properties

Label 1-4024-4024.421-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.973 - 0.227i$
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  + (−0.313 + 0.949i)3-s + (0.686 − 0.726i)5-s + (0.659 − 0.752i)7-s + (−0.803 − 0.595i)9-s + (0.925 + 0.378i)11-s + (−0.939 + 0.343i)13-s + (0.474 + 0.880i)15-s + (0.580 − 0.814i)17-s + (−0.384 + 0.923i)19-s + (0.507 + 0.861i)21-s + (−0.787 + 0.615i)23-s + (−0.0562 − 0.998i)25-s + (0.817 − 0.575i)27-s + (0.668 − 0.743i)29-s + (−0.996 + 0.0875i)31-s + ⋯
L(s)  = 1  + (−0.313 + 0.949i)3-s + (0.686 − 0.726i)5-s + (0.659 − 0.752i)7-s + (−0.803 − 0.595i)9-s + (0.925 + 0.378i)11-s + (−0.939 + 0.343i)13-s + (0.474 + 0.880i)15-s + (0.580 − 0.814i)17-s + (−0.384 + 0.923i)19-s + (0.507 + 0.861i)21-s + (−0.787 + 0.615i)23-s + (−0.0562 − 0.998i)25-s + (0.817 − 0.575i)27-s + (0.668 − 0.743i)29-s + (−0.996 + 0.0875i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.973 - 0.227i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.973 - 0.227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.899966841 - 0.2193677075i\)
\(L(\frac12)\) \(\approx\) \(1.899966841 - 0.2193677075i\)
\(L(1)\) \(\approx\) \(1.179459278 + 0.05932711264i\)
\(L(1)\) \(\approx\) \(1.179459278 + 0.05932711264i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.313 + 0.949i)T \)
5 \( 1 + (0.686 - 0.726i)T \)
7 \( 1 + (0.659 - 0.752i)T \)
11 \( 1 + (0.925 + 0.378i)T \)
13 \( 1 + (-0.939 + 0.343i)T \)
17 \( 1 + (0.580 - 0.814i)T \)
19 \( 1 + (-0.384 + 0.923i)T \)
23 \( 1 + (-0.787 + 0.615i)T \)
29 \( 1 + (0.668 - 0.743i)T \)
31 \( 1 + (-0.996 + 0.0875i)T \)
37 \( 1 + (0.971 - 0.235i)T \)
41 \( 1 + (0.900 + 0.435i)T \)
43 \( 1 + (0.958 - 0.283i)T \)
47 \( 1 + (0.600 + 0.799i)T \)
53 \( 1 + (-0.384 - 0.923i)T \)
59 \( 1 + (-0.955 + 0.295i)T \)
61 \( 1 + (0.649 + 0.760i)T \)
67 \( 1 + (-0.474 + 0.880i)T \)
71 \( 1 + (0.192 - 0.981i)T \)
73 \( 1 + (-0.485 - 0.874i)T \)
79 \( 1 + (-0.180 - 0.983i)T \)
83 \( 1 + (0.155 + 0.987i)T \)
89 \( 1 + (0.539 - 0.842i)T \)
97 \( 1 + (0.889 + 0.457i)T \)
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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

−18.39357858105118104819287335490, −17.89883678321143323738899892045, −17.193459611149075373266237839397, −16.89572445460322905773963665431, −15.723216694225070070641655980363, −14.723445280114274265779748696974, −14.43510302489834088521199297779, −13.90817516792796950378288361732, −12.791246228126587090523922641416, −12.46816849740397996181350006900, −11.64382300243498208063251271568, −11.006409131132656293973261342451, −10.41852844902288721193556833958, −9.36939915126812638981146329258, −8.75278812397889283841410272913, −7.90675815900453164904379690242, −7.2581986851471126913798216515, −6.43519291311709537464222371372, −5.93158324941779962708879819809, −5.31388888596179556220590928933, −4.34917807918155629933277335542, −3.10903075634939130433265475124, −2.39738468599043282632945642653, −1.8489849251656708512171195253, −0.903211472754644794738082376092, 0.671599299660053680171824133282, 1.58339007722047446525194120391, 2.45605303404710986064773208255, 3.68585446962595103399167238551, 4.389249766943732838824088385997, 4.7437079488046891220995954612, 5.69358000381386029515935105993, 6.20689010285822125335059227133, 7.35276830449718998026717382075, 7.96954598843966849194928044552, 9.04764086200006029308170990241, 9.5009204808885968961376488136, 10.04918196835054396586557101061, 10.71899401122478218642890309768, 11.763065239503265469802425230745, 11.99309644573593840932494312121, 12.932771709989505684077718677245, 14.02400223255296683281121864706, 14.33200271571761772586178672308, 14.84858215642895287220944045343, 16.03529390491825619546683066172, 16.46969085234705862519855131132, 17.062404068102220261520019308385, 17.54759292202912697317883242887, 18.08484984448219575819543350384

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