↑

Properties

Label 1-4011-4011.17-r0-0-0
Degree $1$
Conductor $4011$
Sign $-0.199 + 0.979i$
Analytic cond. $18.6270$
Root an. cond. $18.6270$
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 − 0.175i)2-s + (0.938 + 0.345i)4-s + (0.716 + 0.697i)5-s + (−0.863 − 0.504i)8-s + (−0.583 − 0.812i)10-s + (0.998 + 0.0550i)11-s + (−0.115 + 0.993i)13-s + (0.761 + 0.648i)16-s + (0.224 + 0.974i)17-s + (−0.583 + 0.812i)19-s + (0.431 + 0.901i)20-s + (−0.973 − 0.229i)22-s + (0.949 − 0.314i)23-s + (0.0275 + 0.999i)25-s + (0.287 − 0.957i)26-s + ⋯
L(s)  = 1  + (−0.984 − 0.175i)2-s + (0.938 + 0.345i)4-s + (0.716 + 0.697i)5-s + (−0.863 − 0.504i)8-s + (−0.583 − 0.812i)10-s + (0.998 + 0.0550i)11-s + (−0.115 + 0.993i)13-s + (0.761 + 0.648i)16-s + (0.224 + 0.974i)17-s + (−0.583 + 0.812i)19-s + (0.431 + 0.901i)20-s + (−0.973 − 0.229i)22-s + (0.949 − 0.314i)23-s + (0.0275 + 0.999i)25-s + (0.287 − 0.957i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.199 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.199 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $-0.199 + 0.979i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ -0.199 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8325009969 + 1.018895614i\)
\(L(\frac12)\) \(\approx\) \(0.8325009969 + 1.018895614i\)
\(L(1)\) \(\approx\) \(0.8305181313 + 0.2434248557i\)
\(L(1)\) \(\approx\) \(0.8305181313 + 0.2434248557i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (-0.984 - 0.175i)T \)
5 \( 1 + (0.716 + 0.697i)T \)
11 \( 1 + (0.998 + 0.0550i)T \)
13 \( 1 + (-0.115 + 0.993i)T \)
17 \( 1 + (0.224 + 0.974i)T \)
19 \( 1 + (-0.583 + 0.812i)T \)
23 \( 1 + (0.949 - 0.314i)T \)
29 \( 1 + (-0.991 - 0.131i)T \)
31 \( 1 + (-0.904 + 0.426i)T \)
37 \( 1 + (0.904 + 0.426i)T \)
41 \( 1 + (0.789 + 0.614i)T \)
43 \( 1 + (0.701 + 0.712i)T \)
47 \( 1 + (0.159 - 0.987i)T \)
53 \( 1 + (0.360 - 0.932i)T \)
59 \( 1 + (-0.126 - 0.991i)T \)
61 \( 1 + (0.857 + 0.514i)T \)
67 \( 1 + (-0.421 + 0.906i)T \)
71 \( 1 + (0.277 + 0.960i)T \)
73 \( 1 + (0.889 - 0.456i)T \)
79 \( 1 + (-0.381 + 0.924i)T \)
83 \( 1 + (0.746 - 0.665i)T \)
89 \( 1 + (-0.834 + 0.551i)T \)
97 \( 1 + (-0.652 - 0.757i)T \)
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

−18.1243620918949515556997069363, −17.538170051092040136952836134454, −17.0054178775920161667752150817, −16.52449218394187009558008480332, −15.71890156568880317033783765339, −15.00050694037750630035109307443, −14.36431488477129411079169197083, −13.47862942122852206946136851666, −12.71713695242153679433267288305, −12.09884751256608119476355180235, −11.11686700438295695942195599888, −10.736977883904492860033814610173, −9.66121951778963715355323359122, −9.17811445153294068126101483652, −8.89798740500507495027420751362, −7.75838181290078022384918394325, −7.26239509105717184082024736097, −6.34901668747400367524211663054, −5.66153840000129225106120843722, −5.07039227675692393243227840530, −3.9637901558967411283983303689, −2.83457243044860794142291466735, −2.17800529787116099711988511687, −1.168264294132551736415001578616, −0.56078535270114219458986552838, 1.20903293806893852533714900073, 1.81502254915682128361715190993, 2.50308116295085702253798757989, 3.54203610401267790149148592888, 4.12548284638577795314091118868, 5.53949668741598102496952685302, 6.280742370813911580334303160626, 6.757261752683084792469871565365, 7.43538842714480114857219689596, 8.40005362952274196739568185951, 9.06884695451677719160080987548, 9.683066724430371296090933217047, 10.23222629860679526095546894668, 11.22361895788562521426293639154, 11.32962316190906511625214064121, 12.51589710878498695230621613274, 12.96709994890483995664775460137, 14.13505407434852079253675627164, 14.77397036803712059414108286524, 15.03739657590029261940613074182, 16.47036629442165859433412755762, 16.654516785797125070599702459451, 17.34576457790999478350765271046, 17.98721590196052610236873254492, 18.797833500187054702860645495575

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