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Properties

Label 1-4033-4033.288-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.388 + 0.921i$
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 + (−0.973 + 0.230i)3-s + (0.173 − 0.984i)4-s + (0.549 + 0.835i)5-s + (−0.597 + 0.802i)6-s + (−0.0581 − 0.998i)7-s + (−0.5 − 0.866i)8-s + (0.893 − 0.448i)9-s + (0.957 + 0.286i)10-s + (−0.957 + 0.286i)11-s + (0.0581 + 0.998i)12-s + (−0.0581 − 0.998i)13-s + (−0.686 − 0.727i)14-s + (−0.727 − 0.686i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.973 + 0.230i)3-s + (0.173 − 0.984i)4-s + (0.549 + 0.835i)5-s + (−0.597 + 0.802i)6-s + (−0.0581 − 0.998i)7-s + (−0.5 − 0.866i)8-s + (0.893 − 0.448i)9-s + (0.957 + 0.286i)10-s + (−0.957 + 0.286i)11-s + (0.0581 + 0.998i)12-s + (−0.0581 − 0.998i)13-s + (−0.686 − 0.727i)14-s + (−0.727 − 0.686i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.388 + 0.921i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (288, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.388 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1373253249 - 0.2070193108i\)
\(L(\frac12)\) \(\approx\) \(-0.1373253249 - 0.2070193108i\)
\(L(1)\) \(\approx\) \(0.8746066617 - 0.4444754220i\)
\(L(1)\) \(\approx\) \(0.8746066617 - 0.4444754220i\)

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 \( 1 + (-0.973 + 0.230i)T \)
5 \( 1 + (0.549 + 0.835i)T \)
7 \( 1 + (-0.0581 - 0.998i)T \)
11 \( 1 + (-0.957 + 0.286i)T \)
13 \( 1 + (-0.0581 - 0.998i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.230 - 0.973i)T \)
31 \( 1 + (-0.448 - 0.893i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (0.116 + 0.993i)T \)
53 \( 1 + (0.448 - 0.893i)T \)
59 \( 1 + (-0.686 + 0.727i)T \)
61 \( 1 + (0.918 - 0.396i)T \)
67 \( 1 + (-0.549 + 0.835i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (0.286 + 0.957i)T \)
79 \( 1 + (-0.0581 + 0.998i)T \)
83 \( 1 + (-0.973 + 0.230i)T \)
89 \( 1 + (-0.918 + 0.396i)T \)
97 \( 1 + (0.549 + 0.835i)T \)
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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

−18.56453725952504000200294115702, −18.180138208152086260033462458787, −17.38913477630799398337037191752, −16.73044196219050056128919522261, −16.12600723179543616068726603691, −15.89839653530386722950396284365, −14.87365588267713681512647749823, −14.06287762303655073075409885192, −13.41981651132651968827022454109, −12.6370924117497458964131602569, −12.321729198660494741314405563333, −11.74002433798297032817868373980, −10.86699424469146330302469537231, −9.96463436284066397646417099842, −9.05667350995155654144587447676, −8.43173367057640701788457487582, −7.60162519085752818757193760041, −6.83178767041271568351089123822, −5.83099803317442516441927908451, −5.72994560706487024049724526471, −4.984105128182990476936150066580, −4.36583720813037043341139555322, −3.281279189607875497068834763681, −2.17294367508467842778725100464, −1.602545287782643481231694371845, 0.06048731790663005166014366208, 1.05003246340233916557456608126, 2.112520484100038631252576202526, 2.85329025107238472083992466850, 3.81863618820370292074380720484, 4.34338170958579169371067083415, 5.319396866708172261356501898941, 5.837286930384525161673522560630, 6.46294801538495779842960228735, 7.261533280544201327164114785082, 7.98192084032708977832774993028, 9.588242281412653968491565767844, 10.15918551280875593939479176776, 10.433889915807572798991090835075, 11.00324283459625309363637772169, 11.70814138631028387897357554641, 12.80068579990530671022991181138, 12.94394013431342841034082541649, 13.76608660804235052985801303591, 14.632030681631543828753010576830, 15.1330325632743729449870004566, 15.82158401499308810331871508544, 16.7163530489779469922547055735, 17.39858145542465602875587985422, 18.04134870699331564065776556275

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