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Properties

Label 1-4033-4033.583-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.965 - 0.261i$
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  + (−0.766 − 0.642i)2-s + (0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.766 − 0.642i)5-s + (−0.173 − 0.984i)6-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.173 + 0.984i)10-s + (0.173 − 0.984i)11-s + (−0.5 + 0.866i)12-s + (−0.173 + 0.984i)13-s + (−0.173 − 0.984i)14-s + (−0.173 − 0.984i)15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.766 − 0.642i)5-s + (−0.173 − 0.984i)6-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.173 + 0.984i)10-s + (0.173 − 0.984i)11-s + (−0.5 + 0.866i)12-s + (−0.173 + 0.984i)13-s + (−0.173 − 0.984i)14-s + (−0.173 − 0.984i)15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.965 - 0.261i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.965 - 0.261i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005923106296 - 0.04453458068i\)
\(L(\frac12)\) \(\approx\) \(0.005923106296 - 0.04453458068i\)
\(L(1)\) \(\approx\) \(0.7309140283 + 0.01109875281i\)
\(L(1)\) \(\approx\) \(0.7309140283 + 0.01109875281i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (0.173 - 0.984i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.766 + 0.642i)T \)
31 \( 1 + (-0.766 - 0.642i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 - T \)
97 \( 1 + (0.939 + 0.342i)T \)
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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

−18.455195307823734365010117327668, −18.21430668311338538880354337214, −17.6773834157420297139756174523, −16.85219549079062280026532866373, −15.96469247011589134382196272527, −15.2178788555535898509867228410, −14.8116340216644497205920655760, −14.22722480132136548910280860049, −13.674835596650819587005332257506, −12.56110499566032536112477461959, −11.8928215276354510928061924442, −11.103039666764383608865592813754, −10.31107697200894452042287121452, −9.76851754412241772088823898755, −8.86914220458362096487726053195, −7.98560100763029039767202729869, −7.64683329142092052179950365509, −7.2665212387812366255672551695, −6.47921690940729101643025693475, −5.563221513133335439251407831686, −4.48912680727215836865959881669, −3.81819327376296201574985479004, −2.71865713982007422543050287111, −1.94542350665922066413964145087, −1.13086128026512553165475728959, 0.01468485074018432494682341114, 1.508063523166101149474660724596, 2.01581699240814508745234198109, 2.99332875536235509142422590519, 3.823628131951193459380076324142, 4.30487803379808908251243260887, 5.087434145855636388446001738289, 6.22339367755393008610327792009, 7.44579606696382714312103222910, 7.98991201882582949773222801685, 8.60689941481760992865051672235, 9.06185405854732389733586324940, 9.54555547277905160142906979517, 10.749623050334548473406032337826, 11.21877927701882152237646620879, 11.662341336915085901809340989115, 12.64815662188611005581310991723, 13.17582227047938967611228916581, 14.18996324664836761938716875322, 14.78121427796430316438674310608, 15.68585789324081766954964310369, 16.12753885711370978167746280927, 16.755911583853155558639693390431, 17.43802218643630476122059739180, 18.39216298493798865665630509529

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