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Properties

Label 1-4033-4033.1485-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

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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: $\chi_{4033} (1485, \cdot )$
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 \( 1 + iT \)
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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

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

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