↑

Properties

Label 1-4033-4033.2576-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.999 + 0.00544i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.0581 − 0.998i)3-s + (−0.939 − 0.342i)4-s + (−0.802 + 0.597i)5-s + (0.973 + 0.230i)6-s + (0.396 − 0.918i)7-s + (0.5 − 0.866i)8-s + (−0.993 − 0.116i)9-s + (−0.448 − 0.893i)10-s + (0.448 − 0.893i)11-s + (−0.396 + 0.918i)12-s + (−0.396 + 0.918i)13-s + (0.835 + 0.549i)14-s + (0.549 + 0.835i)15-s + (0.766 + 0.642i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.0581 − 0.998i)3-s + (−0.939 − 0.342i)4-s + (−0.802 + 0.597i)5-s + (0.973 + 0.230i)6-s + (0.396 − 0.918i)7-s + (0.5 − 0.866i)8-s + (−0.993 − 0.116i)9-s + (−0.448 − 0.893i)10-s + (0.448 − 0.893i)11-s + (−0.396 + 0.918i)12-s + (−0.396 + 0.918i)13-s + (0.835 + 0.549i)14-s + (0.549 + 0.835i)15-s + (0.766 + 0.642i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.999 + 0.00544i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2576, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.999 + 0.00544i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0001358927909 - 0.04992843487i\)
\(L(\frac12)\) \(\approx\) \(0.0001358927909 - 0.04992843487i\)
\(L(1)\) \(\approx\) \(0.6782074094 + 0.04556762905i\)
\(L(1)\) \(\approx\) \(0.6782074094 + 0.04556762905i\)

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 \( 1 + (0.0581 - 0.998i)T \)
5 \( 1 + (-0.802 + 0.597i)T \)
7 \( 1 + (0.396 - 0.918i)T \)
11 \( 1 + (0.448 - 0.893i)T \)
13 \( 1 + (-0.396 + 0.918i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.998 - 0.0581i)T \)
31 \( 1 + (0.116 - 0.993i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (-0.727 + 0.686i)T \)
53 \( 1 + (0.116 + 0.993i)T \)
59 \( 1 + (0.835 - 0.549i)T \)
61 \( 1 + (0.957 - 0.286i)T \)
67 \( 1 + (-0.802 - 0.597i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.893 - 0.448i)T \)
79 \( 1 + (-0.396 - 0.918i)T \)
83 \( 1 + (0.0581 - 0.998i)T \)
89 \( 1 + (-0.957 + 0.286i)T \)
97 \( 1 + (-0.802 + 0.597i)T \)
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

−19.07812058056250483558008132246, −18.1354369991861384603506277795, −17.51902763078081951019288032171, −16.7657989799028925867133908281, −16.196924961296706963059781582252, −15.30097813760577318291192314643, −14.66682461108387899383685419042, −14.36178519398873828698983825214, −12.89551781129378750826966968572, −12.52423100166117125411965351915, −11.83679469006213696683753968082, −11.3732332005562808344394389542, −10.45536178765865710209184483317, −9.91028604711591473656781290899, −9.18579293674997547266521626359, −8.5127464363052081921939310672, −8.13169323629728752434088467201, −7.14225177180807561239353042310, −5.530965909754527892121870374463, −5.226753779557982221462823301030, −4.40408744972278872979073402328, −3.87006014113860280868173299075, −2.98322745153966537747107092929, −2.30143646022944598492830978949, −1.23417668382307018543842521613, 0.018266490855355946561449442678, 0.989186339706803086726128798797, 1.85498162493565005086179583606, 3.19843957812310451259007101040, 3.89230833906621745901214752297, 4.58798211436798878588127776584, 5.74073138712555167079932117403, 6.357561505736464992568245713685, 7.03234066333560404302444557711, 7.57377197567604975473585749734, 8.04120580193503377964060627688, 8.80477228950045689379424662797, 9.59340492372475047263168172379, 10.68143935169101543287542927933, 11.28429609085114484914662251935, 11.8833056820043407774775451089, 12.9436192212860761956589445612, 13.498631366785118904196015749256, 14.20995431028759644113113447498, 14.64358223740657818969971190950, 15.21927748797550620035845028719, 16.33252281837283659644478927845, 16.84350051617173947979101714674, 17.32471922081468538107192091062, 18.0598983802941056825899717490

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