Properties

Label 1-4033-4033.263-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.295 + 0.955i$
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 + (0.766 + 0.642i)3-s + (0.766 + 0.642i)4-s + (0.939 − 0.342i)5-s + (0.5 + 0.866i)6-s + (−0.939 + 0.342i)7-s + (0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + 10-s + 11-s + (0.173 + 0.984i)12-s + (−0.173 + 0.984i)13-s − 14-s + (0.939 + 0.342i)15-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)3-s + (0.766 + 0.642i)4-s + (0.939 − 0.342i)5-s + (0.5 + 0.866i)6-s + (−0.939 + 0.342i)7-s + (0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + 10-s + 11-s + (0.173 + 0.984i)12-s + (−0.173 + 0.984i)13-s − 14-s + (0.939 + 0.342i)15-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.295 + 0.955i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.295 + 0.955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.217892471 + 4.363594621i\)
\(L(\frac12)\) \(\approx\) \(3.217892471 + 4.363594621i\)
\(L(1)\) \(\approx\) \(2.341706701 + 1.461263928i\)
\(L(1)\) \(\approx\) \(2.341706701 + 1.461263928i\)

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 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.939 - 0.342i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 - T \)
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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

−18.31397556289032142705885820223, −17.759967164433937772412336859150, −16.93322531565909061770708742693, −15.94868667208756450796424719445, −15.386213495613460438328802906488, −14.534236350989455542993837873410, −13.97736716961288772344159312539, −13.60926506156209301394810612214, −12.886020314548624090122892158650, −12.42497610887988974071754996335, −11.57829232309298777277608080837, −10.71048447871318472422954834180, −9.829687757110196439044705773664, −9.53199222507943370239951081819, −8.65848783568654397306780041986, −7.2945568718556946109117674251, −7.01583200144560198831976103785, −6.30127466600621088165286679641, −5.65204166588909106893675831736, −4.73199158190734689090148058482, −3.56785013745937871414046488345, −3.14863653201087854691867432407, −2.58351856865185100562232673270, −1.53621892689043423810804733093, −0.96224128333096827260433673028, 1.42652052896095475220006081471, 2.41157161056523680204560704492, 2.74971883813039173836134026694, 3.882432100084056754245075158421, 4.31388604532379814592296806653, 5.12412386717423385652854376570, 5.997041398140727292269890754423, 6.59073841108502989683938220897, 7.19118207928732870700887825413, 8.5807729313030046566451621614, 8.763183260743298643426662022284, 9.74999856992391867685749976063, 10.10033829871360661889033263133, 11.33228352866717487057059499157, 11.865071858670148910720649112824, 12.91833325447891453928627396272, 13.30622691118407040388907316403, 13.89382282237652250983271165765, 14.59700594356035916805492381749, 15.09114185053655472274090978366, 15.93105076128159619931207512951, 16.52959661260273784034741946557, 16.90454336939396230485526990885, 17.77239536675643011680945221194, 18.90246729559902648412301968945

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