↑

Properties

Label 1-4033-4033.1064-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.527 - 0.849i$
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  + (0.766 − 0.642i)2-s + (−0.0581 + 0.998i)3-s + (0.173 − 0.984i)4-s + (0.993 − 0.116i)5-s + (0.597 + 0.802i)6-s + (0.597 + 0.802i)7-s + (−0.5 − 0.866i)8-s + (−0.993 − 0.116i)9-s + (0.686 − 0.727i)10-s + (0.686 + 0.727i)11-s + (0.973 + 0.230i)12-s + (0.396 − 0.918i)13-s + (0.973 + 0.230i)14-s + (0.0581 + 0.998i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.0581 + 0.998i)3-s + (0.173 − 0.984i)4-s + (0.993 − 0.116i)5-s + (0.597 + 0.802i)6-s + (0.597 + 0.802i)7-s + (−0.5 − 0.866i)8-s + (−0.993 − 0.116i)9-s + (0.686 − 0.727i)10-s + (0.686 + 0.727i)11-s + (0.973 + 0.230i)12-s + (0.396 − 0.918i)13-s + (0.973 + 0.230i)14-s + (0.0581 + 0.998i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.527 - 0.849i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1064, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.527 - 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.122956038 - 1.736401964i\)
\(L(\frac12)\) \(\approx\) \(3.122956038 - 1.736401964i\)
\(L(1)\) \(\approx\) \(1.932149400 - 0.4370862297i\)
\(L(1)\) \(\approx\) \(1.932149400 - 0.4370862297i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (-0.0581 + 0.998i)T \)
5 \( 1 + (0.993 - 0.116i)T \)
7 \( 1 + (0.597 + 0.802i)T \)
11 \( 1 + (0.686 + 0.727i)T \)
13 \( 1 + (0.396 - 0.918i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.686 - 0.727i)T \)
31 \( 1 + (-0.893 - 0.448i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.396 - 0.918i)T \)
53 \( 1 + (-0.973 - 0.230i)T \)
59 \( 1 + (-0.0581 - 0.998i)T \)
61 \( 1 + (0.686 - 0.727i)T \)
67 \( 1 + (0.286 - 0.957i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (-0.286 + 0.957i)T \)
79 \( 1 + (-0.686 - 0.727i)T \)
83 \( 1 + (-0.993 + 0.116i)T \)
89 \( 1 + (-0.893 + 0.448i)T \)
97 \( 1 + (0.0581 - 0.998i)T \)
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

−18.3494012477598283885567273074, −17.722804251663477721862193150013, −17.193807239302870090036440729063, −16.58459871153200203614112485578, −16.15776561151161036322397918479, −14.582102594696480430508710432410, −14.26943109571232186590221049046, −14.08228488401974732069790591451, −13.261037547742513312449320650388, −12.60476848916039728615745884336, −11.95696351954657353081138649816, −11.13682818515036235860580908793, −10.57939233966599490799948712966, −9.33772511382967652396126310054, −8.56824735667943110341692788829, −7.97358143230672936978487849991, −7.1876967815670270993471604409, −6.5253068777572544334551100334, −5.948996701759355025719189834125, −5.48945202360567055354505289896, −4.31546288170095254462907972705, −3.67465234841037223028748815687, −2.734618368939599878437688605957, −1.65791941081830908457037628762, −1.32077957306917902493817462907, 0.73291099835342497331096933156, 1.904012739644794906719039703230, 2.476097982337174982772532591526, 3.23073379037625848336782033627, 4.24018100804933618111584801473, 4.79226240679749665685274941289, 5.55376015125449332463668690276, 5.903457689339374227684489817120, 6.78651630441867502241827295302, 8.14458585128188076893092688109, 8.93525921970589376644894802711, 9.77365817465497502390280970617, 9.85374649099834767077464130126, 10.97111546706372685652314201759, 11.31853206109359425465077294352, 12.304337450682142198359676857388, 12.69629403605739282762761937593, 13.75265048259323713257463853931, 14.34234810086674979678802901209, 14.76789599021543375976690673817, 15.519645805533035245850352183862, 16.037412662553080476152100879543, 17.09520483847584296481377906809, 17.726070190650160552458371632233, 18.24865808913952246929878463935

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