Properties

Label 1-4033-4033.1227-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.942 - 0.334i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (0.984 + 0.173i)2-s + (−0.597 − 0.802i)3-s + (0.939 + 0.342i)4-s + (−0.727 − 0.686i)5-s + (−0.448 − 0.893i)6-s + (0.973 + 0.230i)7-s + (0.866 + 0.5i)8-s + (−0.286 + 0.957i)9-s + (−0.597 − 0.802i)10-s + (0.597 − 0.802i)11-s + (−0.286 − 0.957i)12-s + (0.230 − 0.973i)13-s + (0.918 + 0.396i)14-s + (−0.116 + 0.993i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (−0.597 − 0.802i)3-s + (0.939 + 0.342i)4-s + (−0.727 − 0.686i)5-s + (−0.448 − 0.893i)6-s + (0.973 + 0.230i)7-s + (0.866 + 0.5i)8-s + (−0.286 + 0.957i)9-s + (−0.597 − 0.802i)10-s + (0.597 − 0.802i)11-s + (−0.286 − 0.957i)12-s + (0.230 − 0.973i)13-s + (0.918 + 0.396i)14-s + (−0.116 + 0.993i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.942 - 0.334i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.942 - 0.334i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.687604379 - 0.8072215903i\)
\(L(\frac12)\) \(\approx\) \(4.687604379 - 0.8072215903i\)
\(L(1)\) \(\approx\) \(1.838605344 - 0.3496792095i\)
\(L(1)\) \(\approx\) \(1.838605344 - 0.3496792095i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.984 + 0.173i)T \)
3 \( 1 + (-0.597 - 0.802i)T \)
5 \( 1 + (-0.727 - 0.686i)T \)
7 \( 1 + (0.973 + 0.230i)T \)
11 \( 1 + (0.597 - 0.802i)T \)
13 \( 1 + (0.230 - 0.973i)T \)
17 \( 1 + (0.642 - 0.766i)T \)
19 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (0.116 + 0.993i)T \)
31 \( 1 + (-0.230 - 0.973i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (0.0581 + 0.998i)T \)
53 \( 1 + (0.686 - 0.727i)T \)
59 \( 1 + (-0.116 + 0.993i)T \)
61 \( 1 + (0.998 + 0.0581i)T \)
67 \( 1 + (0.973 - 0.230i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (-0.597 + 0.802i)T \)
79 \( 1 + (0.957 + 0.286i)T \)
83 \( 1 + (0.396 - 0.918i)T \)
89 \( 1 + (-0.448 - 0.893i)T \)
97 \( 1 + (0.957 - 0.286i)T \)
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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

−18.27971390300425794890420164524, −17.55336287478511625814886636905, −16.756279164918699403029674707857, −16.22321599011727737992285704331, −15.25670475820289631556722314617, −15.069062219512145464378345732324, −14.296001613448284934414360423193, −13.91384892889792772706477224864, −12.595370738854170843729969635051, −11.93625748059925782424064162782, −11.5938155742546050845325778672, −10.897691342627719033909448337136, −10.4043185508282204732686222630, −9.61751426403793168246138955885, −8.55554074936545299619692014152, −7.640347672307740976222535913445, −6.77585308027781195410968212594, −6.438922118190496562834538911969, −5.355903202926053283675926210427, −4.6313362033656126657900984732, −4.12970942546056654696602795633, −3.647690930289355212293167857631, −2.58210795318246828066055257069, −1.66552508869228735591721505865, −0.64099213806784562005278868896, 0.87584961763324948473928106385, 1.257667324751449108730577088290, 2.30135152713287988467573087743, 3.367662751762732509367602447423, 3.99710473713943864524465593327, 5.05130170807380774884519040892, 5.497215851200579457418980368453, 5.94075308859512730272823724183, 7.11025039436535238689282593871, 7.7373150654364469499415529917, 8.08773312872987128729795416068, 8.935848819224208763674656491919, 10.32313644799779879497940976484, 11.178575037583538113375865175361, 11.62397866698199362957485037751, 12.06093103078746522917700971757, 12.74035890418765704840169091509, 13.3961381615528100508114337511, 14.19111426405796416597933847802, 14.60789179706525476366286950782, 15.732704083322279971958435058650, 16.07398483359423826737772624212, 16.888859108834773622040312579593, 17.35582764252386181690103524471, 18.26159313659250660410907377904

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