Properties

Label 1-4033-4033.3249-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.999 - 0.0404i$
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  + (−0.939 + 0.342i)2-s + (−0.835 + 0.549i)3-s + (0.766 − 0.642i)4-s + (−0.396 − 0.918i)5-s + (0.597 − 0.802i)6-s + (−0.993 − 0.116i)7-s + (−0.5 + 0.866i)8-s + (0.396 − 0.918i)9-s + (0.686 + 0.727i)10-s + (0.686 − 0.727i)11-s + (−0.286 + 0.957i)12-s + (0.597 − 0.802i)13-s + (0.973 − 0.230i)14-s + (0.835 + 0.549i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (−0.835 + 0.549i)3-s + (0.766 − 0.642i)4-s + (−0.396 − 0.918i)5-s + (0.597 − 0.802i)6-s + (−0.993 − 0.116i)7-s + (−0.5 + 0.866i)8-s + (0.396 − 0.918i)9-s + (0.686 + 0.727i)10-s + (0.686 − 0.727i)11-s + (−0.286 + 0.957i)12-s + (0.597 − 0.802i)13-s + (0.973 − 0.230i)14-s + (0.835 + 0.549i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.999 - 0.0404i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.999 - 0.0404i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003007295268 - 0.1485164313i\)
\(L(\frac12)\) \(\approx\) \(0.003007295268 - 0.1485164313i\)
\(L(1)\) \(\approx\) \(0.4259086811 + 0.02248847054i\)
\(L(1)\) \(\approx\) \(0.4259086811 + 0.02248847054i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.939 + 0.342i)T \)
3 \( 1 + (-0.835 + 0.549i)T \)
5 \( 1 + (-0.396 - 0.918i)T \)
7 \( 1 + (-0.993 - 0.116i)T \)
11 \( 1 + (0.686 - 0.727i)T \)
13 \( 1 + (0.597 - 0.802i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (0.686 + 0.727i)T \)
31 \( 1 + (-0.893 + 0.448i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (-0.396 + 0.918i)T \)
53 \( 1 + (0.286 - 0.957i)T \)
59 \( 1 + (-0.835 - 0.549i)T \)
61 \( 1 + (0.286 - 0.957i)T \)
67 \( 1 + (0.686 - 0.727i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (-0.286 - 0.957i)T \)
79 \( 1 + (0.973 + 0.230i)T \)
83 \( 1 + (0.597 - 0.802i)T \)
89 \( 1 + (0.835 - 0.549i)T \)
97 \( 1 + (0.0581 + 0.998i)T \)
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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

−18.57905630118922262967038394573, −18.287987431972351092602115418814, −17.65385904097052561701995122884, −16.879309029480778899105335490456, −16.11122778549986581074327192923, −15.83213879922258389215693064520, −14.94180920953191828434566710674, −13.8055241689074998811403852884, −13.26093157426460623703554808526, −12.13577125655754847455934649555, −11.94684056062421390989872729030, −11.279347860672870375146159542159, −10.56269258781896610525241114175, −9.880084248414219549377284374206, −9.29355567106135170143740804643, −8.35093963302038992018906427805, −7.387740281930792132999560757525, −6.95571611561682938579914447232, −6.44299563529292276770687195522, −5.83583465809618252219747099152, −4.309105730404795592810073074466, −3.7597232411492383387328616591, −2.642236927743266994595413294192, −2.081016244212767550991754518416, −1.0604948113193582439079423699, 0.09539466973929645808739266606, 0.840165578714346041099972550586, 1.6485141457160733090690363698, 3.2543949516083338162753930790, 3.70190573411263702625587978977, 4.829462674144216410440694294873, 5.59710162699131944448112900069, 6.16992564055898315736485490970, 6.780809242076458015521812579757, 7.73446311848335712191381244181, 8.55897640762320811109199721085, 9.21520444552363418154001367271, 9.62020562434523278250363344348, 10.59258832632896324478756976303, 11.01577541828753877368773334859, 11.87804859826832351699385947102, 12.38632384491982084671500320675, 13.2517052809712181419975275811, 14.14202840466624043492774570334, 15.22029391766527575818693613009, 15.85896912361226856396332711176, 16.16523317632590327189734671828, 16.564400626653502444039138284294, 17.55409288966173546603828078844, 17.76724461605760716523943659390

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