↑

Properties

Label 1-4033-4033.1098-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.834 + 0.551i$
Analytic cond. $433.406$
Root an. cond. $433.406$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.984 − 0.173i)2-s + (−0.939 − 0.342i)3-s + (0.939 + 0.342i)4-s + (−0.173 − 0.984i)5-s + (0.866 + 0.5i)6-s + (0.173 + 0.984i)7-s + (−0.866 − 0.5i)8-s + (0.766 + 0.642i)9-s + i·10-s − i·11-s + (−0.766 − 0.642i)12-s + (0.642 + 0.766i)13-s − i·14-s + (−0.173 + 0.984i)15-s + (0.766 + 0.642i)16-s + (0.342 + 0.939i)17-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)2-s + (−0.939 − 0.342i)3-s + (0.939 + 0.342i)4-s + (−0.173 − 0.984i)5-s + (0.866 + 0.5i)6-s + (0.173 + 0.984i)7-s + (−0.866 − 0.5i)8-s + (0.766 + 0.642i)9-s + i·10-s − i·11-s + (−0.766 − 0.642i)12-s + (0.642 + 0.766i)13-s − i·14-s + (−0.173 + 0.984i)15-s + (0.766 + 0.642i)16-s + (0.342 + 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.834 + 0.551i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1098, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.834 + 0.551i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03613235162 - 0.1202646112i\)
\(L(\frac12)\) \(\approx\) \(-0.03613235162 - 0.1202646112i\)
\(L(1)\) \(\approx\) \(0.4630347417 - 0.1227379840i\)
\(L(1)\) \(\approx\) \(0.4630347417 - 0.1227379840i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.984 - 0.173i)T \)
3 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 - iT \)
13 \( 1 + (0.642 + 0.766i)T \)
17 \( 1 + (0.342 + 0.939i)T \)
19 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.342 - 0.939i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.342 - 0.939i)T \)
59 \( 1 + (0.984 + 0.173i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (-0.342 + 0.939i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.342 - 0.939i)T \)
83 \( 1 + (-0.173 + 0.984i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.32052529964546527756024228993, −17.97204427302116439780586226056, −17.536058363590370643371261705822, −16.745472365761065408362979715675, −16.04798607152056228101828720319, −15.618614409054703801056735257629, −14.65387731570391598666528602034, −14.39405273272298019887903080375, −13.04668322996603331732380093217, −12.33870477258527248569537611443, −11.4455740799652798459028770150, −10.95669866496515906306793231702, −10.37693120764039611811407131738, −10.02334679809032657905162412052, −9.188674027880777812481214312586, −8.06504166770416917052207661224, −7.400404336758608590753560432703, −6.8802827096863652185512396154, −6.29917322852434332379231604689, −5.45357788958856096005298959982, −4.50350325442835153216124774446, −3.68429414514789738182614685589, −2.77312117954080938116994756684, −1.70061698116580447225376747110, −0.83971178529748102227425929424, 0.04753578506650261835068906815, 0.74050718638706476294494794744, 1.876536546089541491085310433501, 1.97462680486199065998577681728, 3.6108798862618821943547612291, 4.25132544566813376496797268009, 5.55855835695669427455335214304, 5.9018131108820865016866993965, 6.47644067506010326027172873817, 7.67824463134360144117557651883, 8.21861348264102971368169343089, 8.77773868270476433887439546102, 9.4838940770900223861029912994, 10.32933649169868388962666250167, 11.25347690409207013067663840318, 11.52150364966927178105491729974, 12.226224877478511467788263211579, 12.83280719132834294464377083106, 13.47129982588469941591895298939, 14.727587665291417519740924823552, 15.58042195831679331394603626299, 16.206770709213263814950550307386, 16.542165702403083431930767739873, 17.2966730020427669878084584108, 17.79164660481265698327989753033

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