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Properties

Label 1-4033-4033.2675-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.0803 - 0.996i$
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  + (−0.642 − 0.766i)2-s + (−0.286 + 0.957i)3-s + (−0.173 + 0.984i)4-s + (0.0581 + 0.998i)5-s + (0.918 − 0.396i)6-s + (0.893 + 0.448i)7-s + (0.866 − 0.5i)8-s + (−0.835 − 0.549i)9-s + (0.727 − 0.686i)10-s + (0.727 + 0.686i)11-s + (−0.893 − 0.448i)12-s + (−0.448 + 0.893i)13-s + (−0.230 − 0.973i)14-s + (−0.973 − 0.230i)15-s + (−0.939 − 0.342i)16-s + (−0.342 + 0.939i)17-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)2-s + (−0.286 + 0.957i)3-s + (−0.173 + 0.984i)4-s + (0.0581 + 0.998i)5-s + (0.918 − 0.396i)6-s + (0.893 + 0.448i)7-s + (0.866 − 0.5i)8-s + (−0.835 − 0.549i)9-s + (0.727 − 0.686i)10-s + (0.727 + 0.686i)11-s + (−0.893 − 0.448i)12-s + (−0.448 + 0.893i)13-s + (−0.230 − 0.973i)14-s + (−0.973 − 0.230i)15-s + (−0.939 − 0.342i)16-s + (−0.342 + 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0803 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0803 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.0803 - 0.996i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2675, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.0803 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1633389459 + 0.1506964039i\)
\(L(\frac12)\) \(\approx\) \(-0.1633389459 + 0.1506964039i\)
\(L(1)\) \(\approx\) \(0.6246835063 + 0.2770190839i\)
\(L(1)\) \(\approx\) \(0.6246835063 + 0.2770190839i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.642 - 0.766i)T \)
3 \( 1 + (-0.286 + 0.957i)T \)
5 \( 1 + (0.0581 + 0.998i)T \)
7 \( 1 + (0.893 + 0.448i)T \)
11 \( 1 + (0.727 + 0.686i)T \)
13 \( 1 + (-0.448 + 0.893i)T \)
17 \( 1 + (-0.342 + 0.939i)T \)
19 \( 1 + (0.342 - 0.939i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (-0.286 + 0.957i)T \)
31 \( 1 + (-0.835 - 0.549i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (-0.802 + 0.597i)T \)
53 \( 1 + (-0.549 - 0.835i)T \)
59 \( 1 + (-0.230 + 0.973i)T \)
61 \( 1 + (-0.993 - 0.116i)T \)
67 \( 1 + (0.998 + 0.0581i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (-0.686 + 0.727i)T \)
79 \( 1 + (0.448 + 0.893i)T \)
83 \( 1 + (0.286 - 0.957i)T \)
89 \( 1 + (0.993 + 0.116i)T \)
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

−17.57317329713935333096449821500, −17.218120840995521955622034241622, −16.65088085647307100059220442722, −16.02647335645924112683478812981, −15.07139185117730094330935080249, −14.30727821682270353396306412463, −13.71908293013889927454587740311, −13.22904509424680523157096162254, −12.229964592377034621248214970471, −11.5508399784831349791207985949, −10.99007843121239089074082542897, −10.03164482513088123657922724515, −9.1680528055548047759227851361, −8.58346743791729025001880907539, −7.80127183673117065943126897914, −7.54382850630356928704673335868, −6.62386535405089519401307494605, −5.723759156322855651575407940074, −5.279906380151611934888056558, −4.62832339639593469553349086773, −3.40559662384585685057915977641, −1.98945887717644773459225764520, −1.39055280056475675484074064130, −0.72560951549285561888002613452, −0.05469092911690302921439995210, 1.42686427121975452824034254368, 2.13714900133827734062016239498, 2.92500145558328436898208943112, 3.75275543948441408275158305434, 4.49725660838882928710388579258, 5.033156317282148954904342422977, 6.33414334166867009882755703087, 6.91959294386637678565384454644, 7.77838041342175272022753484558, 8.77170276139318827176523876995, 9.20858059552299985470760409218, 9.87819817526029232472014275034, 10.663067752420816601026865707963, 11.19184470169439202103555936338, 11.60243926600059742324841550670, 12.29281610022745632402056431863, 13.25255923467900054024930113201, 14.39355613399313239153783893954, 14.705755681204145498267742656801, 15.3083316053911553804852554642, 16.29857635666790869764232293271, 17.01644385427741795816786309613, 17.500445632838611558155733771893, 18.11593467017326631070937803115, 18.70965683158429024612245852010

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