Properties

Label 1-4033-4033.1824-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.993 + 0.112i$
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  + (−0.173 + 0.984i)2-s + (0.973 + 0.230i)3-s + (−0.939 − 0.342i)4-s + (0.835 + 0.549i)5-s + (−0.396 + 0.918i)6-s + (−0.0581 + 0.998i)7-s + (0.5 − 0.866i)8-s + (0.893 + 0.448i)9-s + (−0.686 + 0.727i)10-s + (−0.686 − 0.727i)11-s + (−0.835 − 0.549i)12-s + (0.0581 − 0.998i)13-s + (−0.973 − 0.230i)14-s + (0.686 + 0.727i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.973 + 0.230i)3-s + (−0.939 − 0.342i)4-s + (0.835 + 0.549i)5-s + (−0.396 + 0.918i)6-s + (−0.0581 + 0.998i)7-s + (0.5 − 0.866i)8-s + (0.893 + 0.448i)9-s + (−0.686 + 0.727i)10-s + (−0.686 − 0.727i)11-s + (−0.835 − 0.549i)12-s + (0.0581 − 0.998i)13-s + (−0.973 − 0.230i)14-s + (0.686 + 0.727i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.993 + 0.112i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.993 + 0.112i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1161420114 + 2.049698214i\)
\(L(\frac12)\) \(\approx\) \(0.1161420114 + 2.049698214i\)
\(L(1)\) \(\approx\) \(0.9450015311 + 0.9332438575i\)
\(L(1)\) \(\approx\) \(0.9450015311 + 0.9332438575i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (0.973 + 0.230i)T \)
5 \( 1 + (0.835 + 0.549i)T \)
7 \( 1 + (-0.0581 + 0.998i)T \)
11 \( 1 + (-0.686 - 0.727i)T \)
13 \( 1 + (0.0581 - 0.998i)T \)
17 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.286 + 0.957i)T \)
31 \( 1 + (0.835 - 0.549i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.597 - 0.802i)T \)
53 \( 1 + (-0.0581 + 0.998i)T \)
59 \( 1 + (0.686 + 0.727i)T \)
61 \( 1 + (-0.396 + 0.918i)T \)
67 \( 1 + (0.893 + 0.448i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (-0.686 - 0.727i)T \)
79 \( 1 + (0.835 - 0.549i)T \)
83 \( 1 + (-0.686 + 0.727i)T \)
89 \( 1 + (-0.597 - 0.802i)T \)
97 \( 1 + (0.0581 - 0.998i)T \)
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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

−18.130212453501847394793670315553, −17.585829844519632496062964380984, −17.1537742999232673326785232596, −16.14506198581169358286489505076, −15.376971472841032281447060231143, −14.20659169280336468502742893484, −13.96394073316888210654893633609, −13.303936969143453403512191571, −12.78716886502558980350178241010, −12.212552558075875361315601276136, −11.09766951834479717255651179626, −10.37746599635795815494914527221, −9.81399771262042828409361153034, −9.284145750135702546310723397998, −8.50702559311617400402861308458, −7.98369035805961391532056711804, −6.99521953918169323688311171037, −6.339962262791927953058113767935, −4.90249573918101439001130248255, −4.40778401151337660185286894893, −3.8616311198010465934699903242, −2.64175528726232607809099687470, −2.16724297230155291434976081937, −1.52518922631434824840866641323, −0.51873374849558356748668260232, 1.22756947507189651951379973838, 2.328351161796911435791350586431, 2.90606138897853674942703780129, 3.72822722516237412735983644886, 4.819926817780123541192835347889, 5.63252671803976121271894146250, 5.96833513939735776155329922174, 6.97018232938757482951180212368, 7.73090614192854360767252203036, 8.41713968971122516485776608216, 8.88364836045469295084208727111, 9.71297931255531698906359331796, 10.19364216381047768219522029892, 10.879201295525372888237108058212, 12.17215299992229657038586649160, 13.13825571236803644314228729800, 13.4612055885425301227819806971, 14.18231017350555622595518980515, 14.7606818219226214447428729840, 15.488734300586406552468516189592, 15.790068900813387699495337870852, 16.588240406321029800706589653293, 17.57487003848567559990132530290, 18.29356018191318245261444153721, 18.47986449375469278603815822119

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