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Properties

Label 1-4033-4033.2269-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.0332 + 0.999i$
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 + (−0.286 − 0.957i)3-s + (0.766 + 0.642i)4-s + (−0.835 − 0.549i)5-s + (−0.0581 + 0.998i)6-s + (−0.0581 + 0.998i)7-s + (−0.5 − 0.866i)8-s + (−0.835 + 0.549i)9-s + (0.597 + 0.802i)10-s + (0.597 − 0.802i)11-s + (0.396 − 0.918i)12-s + (0.893 − 0.448i)13-s + (0.396 − 0.918i)14-s + (−0.286 + 0.957i)15-s + (0.173 + 0.984i)16-s + (−0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.286 − 0.957i)3-s + (0.766 + 0.642i)4-s + (−0.835 − 0.549i)5-s + (−0.0581 + 0.998i)6-s + (−0.0581 + 0.998i)7-s + (−0.5 − 0.866i)8-s + (−0.835 + 0.549i)9-s + (0.597 + 0.802i)10-s + (0.597 − 0.802i)11-s + (0.396 − 0.918i)12-s + (0.893 − 0.448i)13-s + (0.396 − 0.918i)14-s + (−0.286 + 0.957i)15-s + (0.173 + 0.984i)16-s + (−0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0332 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0332 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.0332 + 0.999i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.0332 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03070353674 + 0.02969814912i\)
\(L(\frac12)\) \(\approx\) \(0.03070353674 + 0.02969814912i\)
\(L(1)\) \(\approx\) \(0.4398494370 - 0.2193380338i\)
\(L(1)\) \(\approx\) \(0.4398494370 - 0.2193380338i\)

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 \( 1 + (-0.286 - 0.957i)T \)
5 \( 1 + (-0.835 - 0.549i)T \)
7 \( 1 + (-0.0581 + 0.998i)T \)
11 \( 1 + (0.597 - 0.802i)T \)
13 \( 1 + (0.893 - 0.448i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.597 + 0.802i)T \)
31 \( 1 + (-0.686 - 0.727i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (0.893 + 0.448i)T \)
53 \( 1 + (0.396 - 0.918i)T \)
59 \( 1 + (-0.286 + 0.957i)T \)
61 \( 1 + (0.597 + 0.802i)T \)
67 \( 1 + (-0.993 - 0.116i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.993 - 0.116i)T \)
79 \( 1 + (0.597 - 0.802i)T \)
83 \( 1 + (-0.835 - 0.549i)T \)
89 \( 1 + (-0.686 + 0.727i)T \)
97 \( 1 + (-0.286 - 0.957i)T \)
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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

−18.1493403197163819114902262813, −17.50690414183984712038778999236, −16.897051448186438706879746330646, −16.29006836071641010865090767402, −15.74820139208235550383693230400, −15.09768244701599389428590089688, −14.47734101093243156214551662553, −13.9620270223779761183950295885, −12.56115559046955659537214856485, −11.80409468408166944961596659175, −11.0757214194928417623469685078, −10.61435294813283289309211289287, −10.19244638964496053808529978946, −9.259582996100864718433391065413, −8.613541827665082851024862768505, −7.95518952041771980982848621206, −6.95082396756930655836900516018, −6.60145072804605783137365638674, −5.84734149000788591749297323700, −4.45913106316308718399554120208, −4.20959589121425268074815939740, −3.35080684251162252603534719874, −2.26106144424013289783322332345, −1.16150885146294005311316950823, −0.022458014935353425943946941915, 0.92915379169791716627092895504, 1.6205016191515113543804186710, 2.64232823876608401225233902198, 3.25446633957611140154180987621, 4.26808566041441627827486368392, 5.472275297904807938310779196209, 6.09562720931933772132185748039, 6.904243842589720203030735770765, 7.54115591386495201547557015879, 8.446529677294982232514832066418, 8.78546028697044461895281007789, 9.19822749722881749820759385775, 10.649846057545556299256295998296, 11.289534084835623185158880854468, 11.628663674475796036607127703942, 12.24047983557070162502551704423, 13.13242615919565021721220227898, 13.353439260189060418323929334690, 14.75772037256152931781360769495, 15.51800024044166218053806508374, 16.07209192245468903324760859778, 16.67094355058689663903604941519, 17.51495437213537086384206239891, 18.02029536123457658349310412036, 18.6551672327994822694236636888

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