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Properties

Label 1-4033-4033.48-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.685 + 0.728i$
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 + (−0.993 + 0.116i)3-s + (0.173 + 0.984i)4-s + (0.286 + 0.957i)5-s + (0.835 + 0.549i)6-s + (−0.686 + 0.727i)7-s + (0.5 − 0.866i)8-s + (0.973 − 0.230i)9-s + (0.396 − 0.918i)10-s + (0.396 + 0.918i)11-s + (−0.286 − 0.957i)12-s + (0.686 − 0.727i)13-s + (0.993 − 0.116i)14-s + (−0.396 − 0.918i)15-s + (−0.939 + 0.342i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.993 + 0.116i)3-s + (0.173 + 0.984i)4-s + (0.286 + 0.957i)5-s + (0.835 + 0.549i)6-s + (−0.686 + 0.727i)7-s + (0.5 − 0.866i)8-s + (0.973 − 0.230i)9-s + (0.396 − 0.918i)10-s + (0.396 + 0.918i)11-s + (−0.286 − 0.957i)12-s + (0.686 − 0.727i)13-s + (0.993 − 0.116i)14-s + (−0.396 − 0.918i)15-s + (−0.939 + 0.342i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.685 + 0.728i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.685 + 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7505020484 + 0.3244901272i\)
\(L(\frac12)\) \(\approx\) \(0.7505020484 + 0.3244901272i\)
\(L(1)\) \(\approx\) \(0.6011407493 + 0.05181777700i\)
\(L(1)\) \(\approx\) \(0.6011407493 + 0.05181777700i\)

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 \( 1 + (-0.993 + 0.116i)T \)
5 \( 1 + (0.286 + 0.957i)T \)
7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (0.396 + 0.918i)T \)
13 \( 1 + (0.686 - 0.727i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.597 - 0.802i)T \)
31 \( 1 + (0.286 - 0.957i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (0.893 + 0.448i)T \)
53 \( 1 + (-0.686 + 0.727i)T \)
59 \( 1 + (-0.396 - 0.918i)T \)
61 \( 1 + (0.835 + 0.549i)T \)
67 \( 1 + (0.973 - 0.230i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (0.396 + 0.918i)T \)
79 \( 1 + (0.286 - 0.957i)T \)
83 \( 1 + (0.396 - 0.918i)T \)
89 \( 1 + (-0.893 + 0.448i)T \)
97 \( 1 + (0.686 - 0.727i)T \)
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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

−18.386217613013054357230499053819, −17.29619158753643292409531001727, −16.953742568966014732397717915757, −16.5132492472041815017486059121, −16.147139752804830013858246633719, −15.32146027324084485300641097297, −14.152102893780041317499007596474, −13.738791078831711007075677665798, −12.831582422492577488086913983928, −12.25828070639068300959367357106, −11.297456011335825207684811996438, −10.64853129591329824572503868301, −10.13177409320100647890727243240, −9.282829488606075180401810504084, −8.67838419299911605772396479608, −7.97856930928945130108276526528, −6.82506088672289304176696120693, −6.6438863881075724706115744600, −5.6769183892554276572674572280, −5.32576612127232896582747171226, −4.25788451991864174619294160595, −3.564321803909688037311216846500, −1.85915287876052901065642654710, −1.17900466659602642673096285315, −0.56824438236847061861235084504, 0.72047232688028167117898061294, 1.74486545613112691174850129423, 2.604821759136470547172529212144, 3.31487672858251184262994474069, 4.11991690869182639519408783992, 5.16365578967348536328421407917, 6.127924372282958750937025239324, 6.55660017047469167300514782092, 7.34338167803388884155724830550, 8.04627657086909480601891882171, 9.361701927194175343649159665984, 9.5729004096008426975582805938, 10.29347375432328055209656231362, 10.9974586686501903756555683734, 11.51282123222517472820701511231, 12.19001232523947898141451567179, 12.96171446917098735492089556679, 13.32214539401300607086328728899, 14.73462927981452947980128677504, 15.4152102672425317804744873614, 15.81815236759353232859780338237, 16.92800182864075117214041493791, 17.24235594548163077767480187948, 17.943878968754577642966504494239, 18.69967309842765080559373397834

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