↑

Properties

Label 1-4033-4033.1696-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.919 + 0.393i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.984 − 0.173i)2-s + (−0.396 + 0.918i)3-s + (0.939 + 0.342i)4-s + (0.230 − 0.973i)5-s + (0.549 − 0.835i)6-s + (−0.286 − 0.957i)7-s + (−0.866 − 0.5i)8-s + (−0.686 − 0.727i)9-s + (−0.396 + 0.918i)10-s + (0.396 + 0.918i)11-s + (−0.686 + 0.727i)12-s + (0.957 − 0.286i)13-s + (0.116 + 0.993i)14-s + (0.802 + 0.597i)15-s + (0.766 + 0.642i)16-s + (−0.642 + 0.766i)17-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)2-s + (−0.396 + 0.918i)3-s + (0.939 + 0.342i)4-s + (0.230 − 0.973i)5-s + (0.549 − 0.835i)6-s + (−0.286 − 0.957i)7-s + (−0.866 − 0.5i)8-s + (−0.686 − 0.727i)9-s + (−0.396 + 0.918i)10-s + (0.396 + 0.918i)11-s + (−0.686 + 0.727i)12-s + (0.957 − 0.286i)13-s + (0.116 + 0.993i)14-s + (0.802 + 0.597i)15-s + (0.766 + 0.642i)16-s + (−0.642 + 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.919 + 0.393i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1696, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.919 + 0.393i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5768912702 + 0.1182446494i\)
\(L(\frac12)\) \(\approx\) \(0.5768912702 + 0.1182446494i\)
\(L(1)\) \(\approx\) \(0.5471231544 - 0.04031786385i\)
\(L(1)\) \(\approx\) \(0.5471231544 - 0.04031786385i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.984 - 0.173i)T \)
3 \( 1 + (-0.396 + 0.918i)T \)
5 \( 1 + (0.230 - 0.973i)T \)
7 \( 1 + (-0.286 - 0.957i)T \)
11 \( 1 + (0.396 + 0.918i)T \)
13 \( 1 + (0.957 - 0.286i)T \)
17 \( 1 + (-0.642 + 0.766i)T \)
19 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (-0.802 + 0.597i)T \)
31 \( 1 + (-0.957 - 0.286i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (-0.893 - 0.448i)T \)
53 \( 1 + (-0.973 - 0.230i)T \)
59 \( 1 + (0.802 + 0.597i)T \)
61 \( 1 + (0.448 + 0.893i)T \)
67 \( 1 + (-0.286 + 0.957i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (-0.396 - 0.918i)T \)
79 \( 1 + (0.727 - 0.686i)T \)
83 \( 1 + (-0.993 + 0.116i)T \)
89 \( 1 + (0.549 - 0.835i)T \)
97 \( 1 + (0.727 + 0.686i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.24743324833044282616969007359, −17.92226354109900078762222631919, −16.9773880672714952256285904620, −16.38734702842653555251278703313, −15.710140204291407865169381231438, −14.93408534270316776769049162665, −14.210403327740092694011464273954, −13.511682325919914083732489205184, −12.71522520354453791270505529150, −11.68461012793918177516624983896, −11.30064524311173622763234952721, −10.95944630665473006616675319766, −9.80486508877399391741575187877, −9.21189294973204953379721294186, −8.39649930701758420354372993481, −7.86109959823025429750523867835, −6.88480134972702361095288036571, −6.426976045800524979469040214237, −5.92213317071346793131853857179, −5.29063659934861655109523858957, −3.47119707683636448996881078866, −2.96389636055288060000647997507, −1.9085486162155155574965740979, −1.56061430961566185535915375157, −0.2488953501632788726603704095, 0.445342902259714175312305094, 1.29972993975173307375121226603, 2.135167828382388605152645877276, 3.42363382685573198645853060052, 4.0129004899811656903297617600, 4.72097989505029474715319937325, 5.66927051048554051601299226487, 6.54513609878526998680213758155, 7.04756690791811798714566789888, 8.20397482931543791506188848653, 8.86004571029246159509859876054, 9.25587739718832784653696900516, 10.13055446706679465213818483827, 10.55993324279614843816420689254, 11.22210369732740720995958428656, 11.94859361524335005745374232104, 12.97604863007624311405389338643, 13.13357369449152962620531973746, 14.58843858131803897412677479817, 15.15434525339508583824521220094, 15.95947880165734694836777495402, 16.446774422324579148924587293167, 17.01180751790053458418745191933, 17.52727717673742356036338832094, 17.99212372112307259099969949238

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