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Properties

Label 1-4033-4033.204-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.471 - 0.881i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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  + (−0.5 − 0.866i)2-s + (0.993 − 0.116i)3-s + (−0.5 + 0.866i)4-s + (−0.727 − 0.686i)5-s + (−0.597 − 0.802i)6-s + (0.973 + 0.230i)7-s + 8-s + (0.973 − 0.230i)9-s + (−0.230 + 0.973i)10-s + (0.230 + 0.973i)11-s + (−0.396 + 0.918i)12-s + (−0.686 + 0.727i)13-s + (−0.286 − 0.957i)14-s + (−0.802 − 0.597i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.993 − 0.116i)3-s + (−0.5 + 0.866i)4-s + (−0.727 − 0.686i)5-s + (−0.597 − 0.802i)6-s + (0.973 + 0.230i)7-s + 8-s + (0.973 − 0.230i)9-s + (−0.230 + 0.973i)10-s + (0.230 + 0.973i)11-s + (−0.396 + 0.918i)12-s + (−0.686 + 0.727i)13-s + (−0.286 − 0.957i)14-s + (−0.802 − 0.597i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.471 - 0.881i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (204, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.471 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.773288223 - 1.062719881i\)
\(L(\frac12)\) \(\approx\) \(1.773288223 - 1.062719881i\)
\(L(1)\) \(\approx\) \(1.107914047 - 0.4927941942i\)
\(L(1)\) \(\approx\) \(1.107914047 - 0.4927941942i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.993 - 0.116i)T \)
5 \( 1 + (-0.727 - 0.686i)T \)
7 \( 1 + (0.973 + 0.230i)T \)
11 \( 1 + (0.230 + 0.973i)T \)
13 \( 1 + (-0.686 + 0.727i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.957 - 0.286i)T \)
31 \( 1 + (-0.448 + 0.893i)T \)
41 \( 1 + (0.642 - 0.766i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (0.802 + 0.597i)T \)
53 \( 1 + (-0.802 + 0.597i)T \)
59 \( 1 + (0.597 - 0.802i)T \)
61 \( 1 + (0.448 + 0.893i)T \)
67 \( 1 + (-0.918 + 0.396i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.286 - 0.957i)T \)
79 \( 1 + (-0.993 - 0.116i)T \)
83 \( 1 + (-0.893 + 0.448i)T \)
89 \( 1 + (0.230 - 0.973i)T \)
97 \( 1 + (0.448 + 0.893i)T \)
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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

−18.71105666369980925614700688831, −17.926341953346799137776380882812, −17.26448732849427497514815808704, −16.41112624837528083345469480749, −15.747270308912957182144306441392, −15.12649939581663257301955668910, −14.5803244620172992234423553710, −14.15990182644182603908376726326, −13.55526077640651497360570120736, −12.51366503107763530290031905849, −11.535430902152849115532389225307, −10.78672283472550110310663218720, −10.1687279518144361447070207916, −9.53078976868108052631443812507, −8.35865570593462556150183984530, −8.24704180449348794162958775201, −7.51396813431417472357800305023, −7.08481980281543039708432167632, −5.91750406402514317949612664819, −5.27013105293911520861913145127, −4.20570151791380561039127665256, −3.6855747865687601545339931867, −2.78166507628531141905132673342, −1.68705629829722301519114373471, −0.83926096385180749061560978324, 0.86447114974761521157526540198, 1.548178190927507253940739272917, 2.39678993174853444511680413910, 2.988896131687622046283852949048, 4.13759536296137936905230833922, 4.549230096564851443090788599519, 5.08377666962927410574382305817, 6.93193251351071060473747325916, 7.494394920368688860807299245091, 7.95154170472499584615929316092, 8.83721056616596039575747759967, 9.1952261104738337734304144530, 9.85470858271937819132548899472, 10.77668572646476643903409088071, 11.667801682190743275989664516173, 12.23081525805254379966736241343, 12.510232067342888707798263467824, 13.56604333361840454891405318833, 14.27667457366643767737660449225, 14.70724009961967343853585041483, 15.814125128765570245057304637262, 16.20066305710013934975778725180, 17.26926163870290667883889188296, 17.80210509199450581479583517337, 18.51848817068878203435864927808

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