↑

Properties

Label 1-4033-4033.2809-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.869 - 0.494i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.939 − 0.342i)2-s + (0.973 − 0.230i)3-s + (0.766 − 0.642i)4-s + (0.893 + 0.448i)5-s + (0.835 − 0.549i)6-s + (−0.835 + 0.549i)7-s + (0.5 − 0.866i)8-s + (0.893 − 0.448i)9-s + (0.993 + 0.116i)10-s + (0.993 − 0.116i)11-s + (0.597 − 0.802i)12-s + (0.0581 + 0.998i)13-s + (−0.597 + 0.802i)14-s + (0.973 + 0.230i)15-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.973 − 0.230i)3-s + (0.766 − 0.642i)4-s + (0.893 + 0.448i)5-s + (0.835 − 0.549i)6-s + (−0.835 + 0.549i)7-s + (0.5 − 0.866i)8-s + (0.893 − 0.448i)9-s + (0.993 + 0.116i)10-s + (0.993 − 0.116i)11-s + (0.597 − 0.802i)12-s + (0.0581 + 0.998i)13-s + (−0.597 + 0.802i)14-s + (0.973 + 0.230i)15-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.869 - 0.494i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.869 - 0.494i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.999851304 - 1.588175187i\)
\(L(\frac12)\) \(\approx\) \(5.999851304 - 1.588175187i\)
\(L(1)\) \(\approx\) \(3.001258194 - 0.6173482611i\)
\(L(1)\) \(\approx\) \(3.001258194 - 0.6173482611i\)

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 \( 1 + (0.973 - 0.230i)T \)
5 \( 1 + (0.893 + 0.448i)T \)
7 \( 1 + (-0.835 + 0.549i)T \)
11 \( 1 + (0.993 - 0.116i)T \)
13 \( 1 + (0.0581 + 0.998i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.993 - 0.116i)T \)
31 \( 1 + (-0.286 - 0.957i)T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.0581 - 0.998i)T \)
53 \( 1 + (-0.597 + 0.802i)T \)
59 \( 1 + (-0.973 - 0.230i)T \)
61 \( 1 + (-0.993 - 0.116i)T \)
67 \( 1 + (-0.396 + 0.918i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (0.396 - 0.918i)T \)
79 \( 1 + (0.993 - 0.116i)T \)
83 \( 1 + (0.893 + 0.448i)T \)
89 \( 1 + (-0.286 + 0.957i)T \)
97 \( 1 + (0.973 - 0.230i)T \)
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

−18.64074648216047892463353912273, −17.62029681183624474887069324861, −16.80904952305316323042731631943, −16.48058769004748680546527110151, −15.75432261087085800557186817213, −14.917294776787447838298779053209, −14.217973196942760681941270171124, −13.9842616633694215493966128783, −13.15137137796776003342602687768, −12.52794677437483026718614250198, −12.23052843581458851864043950733, −10.66132613762517084138833360324, −10.30804172579153774419732608051, −9.46704914449600546941173822298, −8.8072882790119418793452193108, −7.93302782575325721648285437636, −7.30544527458066600937957340613, −6.45349611244172401848193963404, −5.81094627659077677082460633219, −5.047090180306449047811038696825, −4.11411684417070389244825372972, −3.49443764598421183701705806619, −2.97456958893243406280313880420, −1.91484335421616756154009123155, −1.21571639072819487781800594269, 1.24698929019981794897887426087, 1.8409538965614990588872418526, 2.6592981905085950906889438404, 3.26247070512048105821961625590, 3.83893768753471937982571760465, 4.84705293260501835581856290711, 5.85464758336085270148262686014, 6.33952369292019592850620123611, 7.02428408041004488264252085806, 7.67417954047626177021703734714, 9.216684202679540964929840252490, 9.37000464964469171531867879093, 9.858033166897967563235609611430, 10.99383102860295017847492817748, 11.75762086076376220436066031264, 12.318434158943346323181635825515, 13.3551939411353571835508768577, 13.49205613966315273468661118296, 14.20930376885682547609799576811, 14.86073019611742251123192021594, 15.3005018430112474794212928394, 16.304887446956390122424798848072, 16.79435311391766169268253568315, 18.025249213446602189146505099365, 18.71373658213405750858284068829

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