↑

Properties

Label 1-4033-4033.755-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.278 + 0.960i$
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  + (0.766 + 0.642i)2-s + (−0.173 − 0.984i)3-s + (0.173 + 0.984i)4-s + (−0.642 − 0.766i)5-s + (0.5 − 0.866i)6-s + (0.766 − 0.642i)7-s + (−0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s − i·10-s − i·11-s + (0.939 − 0.342i)12-s + (−0.939 − 0.342i)13-s + 14-s + (−0.642 + 0.766i)15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (−0.173 − 0.984i)3-s + (0.173 + 0.984i)4-s + (−0.642 − 0.766i)5-s + (0.5 − 0.866i)6-s + (0.766 − 0.642i)7-s + (−0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s − i·10-s − i·11-s + (0.939 − 0.342i)12-s + (−0.939 − 0.342i)13-s + 14-s + (−0.642 + 0.766i)15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.278 + 0.960i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.278 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.033898290 + 0.7766903423i\)
\(L(\frac12)\) \(\approx\) \(1.033898290 + 0.7766903423i\)
\(L(1)\) \(\approx\) \(1.182749490 + 0.02855227755i\)
\(L(1)\) \(\approx\) \(1.182749490 + 0.02855227755i\)

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 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 + (-0.642 - 0.766i)T \)
7 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 - iT \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.984 + 0.173i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.984 - 0.173i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (-0.342 + 0.939i)T \)
67 \( 1 + (0.984 - 0.173i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.766 - 0.642i)T \)
89 \( 1 + (0.984 + 0.173i)T \)
97 \( 1 - iT \)
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

−18.596093881185460927094197608567, −17.76439882151283984335980779362, −16.883186875678553658349769918935, −15.97457534311261139860045293707, −15.32779557861191026195164364093, −14.89143188291176214638095257191, −14.440348607451605878581826044582, −13.82342523200416106114436789911, −12.50423562606025020803365863034, −11.99430030392753533629626262742, −11.5182019216269260784722625851, −10.91979889327538187593364692940, −10.16462149657979545833641380660, −9.56848706627766812085815093803, −8.91731441399599924894193807118, −7.70306591702058197213632091376, −7.06164622914412121830429932340, −6.12947673692670999426369926392, −5.11096182085955651607500987752, −4.89988674382348772604558047469, −4.03845513442953486528819407289, −3.36366631135684887735000464198, −2.45544291445433541441330931534, −1.99208756991536555698759977839, −0.3065152483111745494916618028, 0.95852050119203696771326257294, 1.767896974959535178388372674671, 2.971569868914942618769393394027, 3.630573311481556797489951337390, 4.56650999109955188678120649519, 5.3377281759583956528376071784, 5.663103640825793951917147940564, 6.884435949864122339307469050194, 7.36006127796890176437795486502, 7.94492956210436276121142767992, 8.49466589354695325077878450479, 9.22541348476091777355391753359, 10.76440993724779266288685414849, 11.40136127816053015792580793557, 11.78165877352960687072562155125, 12.7880888847825078027015626487, 13.02311314290339284832036021333, 13.7883310519004176170781431127, 14.4565556349146210609895753000, 15.07394090355052441411071513771, 15.936587729134850469151523825380, 16.69351226698209214992170022170, 17.14800167217186288400910686807, 17.62100368188934482150880630866, 18.48411075815766480970779798773

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