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Properties

Label 1-4033-4033.3878-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.977 + 0.213i$
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 + (−0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.173 + 0.984i)5-s + (0.5 − 0.866i)6-s + (0.173 − 0.984i)7-s + (0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.173 + 0.984i)12-s + (−0.766 − 0.642i)13-s + (0.5 + 0.866i)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 + (−0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.173 + 0.984i)5-s + (0.5 − 0.866i)6-s + (0.173 − 0.984i)7-s + (0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.173 + 0.984i)12-s + (−0.766 − 0.642i)13-s + (0.5 + 0.866i)14-s + (−0.173 − 0.984i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.977 + 0.213i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3878, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.977 + 0.213i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01020229827 + 0.09463243280i\)
\(L(\frac12)\) \(\approx\) \(0.01020229827 + 0.09463243280i\)
\(L(1)\) \(\approx\) \(0.4146988396 + 0.1167919968i\)
\(L(1)\) \(\approx\) \(0.4146988396 + 0.1167919968i\)

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

−18.39215789019861237181822546119, −17.30502181867964464726810166482, −16.91518875638781025091185295577, −16.54986317136860704944495123643, −15.627010988103909780549325738145, −15.091077606786444847340058715234, −13.6480590087594852294909708325, −12.974523292040623552104732780458, −12.54627441067488966654087658738, −11.82302335998828440229562157545, −11.430335268097369796408921600636, −10.6999353038071007630352850642, −9.82727533110862312691759104647, −9.17013997341920682740730253405, −8.37956148197496472049507309320, −7.94296632282516799583184943929, −7.07321354456271251209152917888, −5.99704731067760053724926484275, −5.55612634707006047846920664583, −4.55978965095252010949828528795, −3.93554657364250109543646846849, −2.686550426559514994497029801779, −1.84529534765758607530439066446, −1.28985710620911550560588961376, −0.06250543816200503911462206452, 0.65722516793986993827606454015, 1.95119485882573468165470080927, 2.81903801482843393958563168394, 4.05719738110164830298028799769, 4.84681268240237856081154095190, 5.31182191885356005652939062392, 6.43126256442760793071550937077, 7.126987526353538347335823302090, 7.175696684179159100795502962311, 8.144673355613943782215776572129, 9.279692302862775335925672163853, 10.01048559932901929495178525289, 10.462716442087476629745213585089, 10.90467983267840858711941537225, 11.60103597002518519175498067582, 12.54755781461470982958593440138, 13.42315673956587273012846964383, 14.34844455804826858014378381435, 14.94715108643836897925006142427, 15.47066066398106311001136669889, 16.08720263533843815196053545800, 16.99885774534572374097792794120, 17.37380310584747791706433596735, 17.919602329082437468363231280509, 18.53815642786091808507461079218

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