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Properties

Label 1-4033-4033.35-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.547 + 0.836i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + (−0.893 − 0.448i)3-s − 4-s + (0.116 − 0.993i)5-s + (−0.448 + 0.893i)6-s + (0.597 − 0.802i)7-s + i·8-s + (0.597 + 0.802i)9-s + (−0.993 − 0.116i)10-s + (0.993 − 0.116i)11-s + (0.893 + 0.448i)12-s + (0.116 − 0.993i)13-s + (−0.802 − 0.597i)14-s + (−0.549 + 0.835i)15-s + 16-s + (−0.866 − 0.5i)17-s + ⋯
L(s)  = 1  − i·2-s + (−0.893 − 0.448i)3-s − 4-s + (0.116 − 0.993i)5-s + (−0.448 + 0.893i)6-s + (0.597 − 0.802i)7-s + i·8-s + (0.597 + 0.802i)9-s + (−0.993 − 0.116i)10-s + (0.993 − 0.116i)11-s + (0.893 + 0.448i)12-s + (0.116 − 0.993i)13-s + (−0.802 − 0.597i)14-s + (−0.549 + 0.835i)15-s + 16-s + (−0.866 − 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.547 + 0.836i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.547 + 0.836i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.018152967 - 0.5507377181i\)
\(L(\frac12)\) \(\approx\) \(-1.018152967 - 0.5507377181i\)
\(L(1)\) \(\approx\) \(0.3663269825 - 0.7120574663i\)
\(L(1)\) \(\approx\) \(0.3663269825 - 0.7120574663i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 + (-0.893 - 0.448i)T \)
5 \( 1 + (0.116 - 0.993i)T \)
7 \( 1 + (0.597 - 0.802i)T \)
11 \( 1 + (0.993 - 0.116i)T \)
13 \( 1 + (0.116 - 0.993i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.984 + 0.173i)T \)
23 \( 1 + (-0.642 + 0.766i)T \)
29 \( 1 + (-0.802 - 0.597i)T \)
31 \( 1 + (0.230 - 0.973i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.984 + 0.173i)T \)
47 \( 1 + (0.893 - 0.448i)T \)
53 \( 1 + (-0.0581 - 0.998i)T \)
59 \( 1 + (0.549 - 0.835i)T \)
61 \( 1 + (0.957 + 0.286i)T \)
67 \( 1 + (-0.893 + 0.448i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.597 - 0.802i)T \)
79 \( 1 + (0.549 - 0.835i)T \)
83 \( 1 + (-0.686 - 0.727i)T \)
89 \( 1 + (0.918 - 0.396i)T \)
97 \( 1 + (0.230 + 0.973i)T \)
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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

−18.455163861551458356016750907666, −18.12175699469741952045313769079, −17.45025352918878015236669798378, −16.87047644081294011100716224777, −16.115338844744569278033102735811, −15.443013324061843640148680065915, −14.97063136200361719373010573537, −14.24859326838655445808280588114, −13.82339586570573809741580465569, −12.6099004097638510904662226193, −11.91578227129953873660873071344, −11.37167993512376950514182753841, −10.57910874851916184722201073798, −9.80338894650072061221522977786, −9.06599017513103270608786824941, −8.58943961171287174429806114930, −7.35068045427164460072923504251, −6.797445107027682734740333090363, −6.27401924032276197123308913969, −5.61502764667883190672579550461, −4.82696069409767918799258018794, −4.10739005585962555803666545941, −3.441519994245698101829952353643, −2.07865456631951329327171798459, −1.16528851189170294950439341437, 0.28249456673315291083837490486, 0.7145690845436304723547774222, 1.51362285515956002084543365083, 2.05431198688099701211181271105, 3.495828754758716688986184714262, 4.15766830812054061955117745654, 4.89652938909965978553610645774, 5.438336689164997070783310574731, 6.22537932296427607275030976268, 7.40343036616092819203713256593, 7.96793974555830922896180783721, 8.73297375636456656174008133348, 9.73704779902957570020094814232, 10.07165659664824153632295543416, 11.1283858629027820938886115907, 11.67006019519018578847597708837, 11.904339100886744078937328404382, 13.022035443587800745328854056026, 13.37123332155867410233812954245, 13.84693174723982984923827227612, 14.88627578910252438849768042951, 15.91199733185564360684677356477, 16.72061707166153217853531956704, 17.23664419026429928776071476873, 17.74350282580434808978095997595

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