Properties

Label 1-4033-4033.3061-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.283 + 0.958i$
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  + (0.939 + 0.342i)2-s + (−0.835 + 0.549i)3-s + (0.766 + 0.642i)4-s + (0.993 + 0.116i)5-s + (−0.973 + 0.230i)6-s + (0.597 − 0.802i)7-s + (0.5 + 0.866i)8-s + (0.396 − 0.918i)9-s + (0.893 + 0.448i)10-s + (0.893 − 0.448i)11-s + (−0.993 − 0.116i)12-s + (−0.597 + 0.802i)13-s + (0.835 − 0.549i)14-s + (−0.893 + 0.448i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (−0.835 + 0.549i)3-s + (0.766 + 0.642i)4-s + (0.993 + 0.116i)5-s + (−0.973 + 0.230i)6-s + (0.597 − 0.802i)7-s + (0.5 + 0.866i)8-s + (0.396 − 0.918i)9-s + (0.893 + 0.448i)10-s + (0.893 − 0.448i)11-s + (−0.993 − 0.116i)12-s + (−0.597 + 0.802i)13-s + (0.835 − 0.549i)14-s + (−0.893 + 0.448i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.283 + 0.958i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3061, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.283 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.985335505 + 2.230410823i\)
\(L(\frac12)\) \(\approx\) \(2.985335505 + 2.230410823i\)
\(L(1)\) \(\approx\) \(1.844220358 + 0.7732023360i\)
\(L(1)\) \(\approx\) \(1.844220358 + 0.7732023360i\)

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 \( 1 + (-0.835 + 0.549i)T \)
5 \( 1 + (0.993 + 0.116i)T \)
7 \( 1 + (0.597 - 0.802i)T \)
11 \( 1 + (0.893 - 0.448i)T \)
13 \( 1 + (-0.597 + 0.802i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.0581 + 0.998i)T \)
31 \( 1 + (0.993 - 0.116i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (-0.686 + 0.727i)T \)
53 \( 1 + (0.597 - 0.802i)T \)
59 \( 1 + (-0.893 + 0.448i)T \)
61 \( 1 + (-0.973 + 0.230i)T \)
67 \( 1 + (0.396 - 0.918i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.893 - 0.448i)T \)
79 \( 1 + (0.993 - 0.116i)T \)
83 \( 1 + (0.893 + 0.448i)T \)
89 \( 1 + (0.686 + 0.727i)T \)
97 \( 1 + (-0.597 + 0.802i)T \)
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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

−18.28204194758502589879363766458, −17.606236904647358853939656781592, −17.09747529554943937637557945610, −16.38771542332458198357336366429, −15.28275319335035620762996120674, −14.95774416581407897893054051556, −14.037768163972057041363267490546, −13.49847656275560619290706027691, −12.737941273674591808959131682181, −12.16306466765195739327799336636, −11.77938351847572501371700711765, −10.911886759721275692293884433907, −10.18093692844339960624475577029, −9.672361981921943027770535347416, −8.53224464862447728593482746848, −7.66330342238982325876482614429, −6.65774774066857338724266408511, −6.238033836918598813674392867351, −5.56812860097789837552177428536, −4.91825003169561029466076326611, −4.41755252281275935623267945367, −3.04837902292093887928447202956, −2.23023842891665026905691476567, −1.70526459340940983511931701170, −0.897962106597933265052303399023, 1.14426982170156944094862924612, 1.79745385325058588270504024466, 3.07116848912829735228782539054, 3.724464972535009017035783178059, 4.67913739831430082195433266309, 5.03747622145743472157646665007, 5.77963576964972859180562176505, 6.66399841043722598841036900685, 6.937842665848049986117883865758, 7.887320784349397914293068569123, 9.138520108641238703750208661532, 9.64580417306406064183368962401, 10.52232974201398713307708558592, 11.20189231781785404414323940788, 11.79812571594994342293867440493, 12.25514784369854924384744363949, 13.549041596035528743092532098284, 13.77107992146634290992924970283, 14.450320660889680019292228117873, 15.04452546911252108811306768368, 16.09344687690125842762145663620, 16.56552381028151277476656527252, 17.03203212741139510886735874378, 17.70101170022926573688933657602, 18.19510877402173176459944134781

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