Properties

Label 1-4011-4011.206-r0-0-0
Degree $1$
Conductor $4011$
Sign $-0.961 + 0.274i$
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  + (−0.159 + 0.987i)2-s + (−0.949 − 0.314i)4-s + (0.993 + 0.110i)5-s + (0.461 − 0.887i)8-s + (−0.266 + 0.963i)10-s + (−0.904 + 0.426i)11-s + (−0.601 + 0.799i)13-s + (0.802 + 0.596i)16-s + (−0.234 + 0.972i)17-s + (−0.266 − 0.963i)19-s + (−0.909 − 0.416i)20-s + (−0.277 − 0.960i)22-s + (0.834 − 0.551i)23-s + (0.975 + 0.218i)25-s + (−0.693 − 0.720i)26-s + ⋯
L(s)  = 1  + (−0.159 + 0.987i)2-s + (−0.949 − 0.314i)4-s + (0.993 + 0.110i)5-s + (0.461 − 0.887i)8-s + (−0.266 + 0.963i)10-s + (−0.904 + 0.426i)11-s + (−0.601 + 0.799i)13-s + (0.802 + 0.596i)16-s + (−0.234 + 0.972i)17-s + (−0.266 − 0.963i)19-s + (−0.909 − 0.416i)20-s + (−0.277 − 0.960i)22-s + (0.834 − 0.551i)23-s + (0.975 + 0.218i)25-s + (−0.693 − 0.720i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $-0.961 + 0.274i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ -0.961 + 0.274i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1729255106 + 1.235582455i\)
\(L(\frac12)\) \(\approx\) \(0.1729255106 + 1.235582455i\)
\(L(1)\) \(\approx\) \(0.7729004247 + 0.5653283778i\)
\(L(1)\) \(\approx\) \(0.7729004247 + 0.5653283778i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (-0.159 + 0.987i)T \)
5 \( 1 + (0.993 + 0.110i)T \)
11 \( 1 + (-0.904 + 0.426i)T \)
13 \( 1 + (-0.601 + 0.799i)T \)
17 \( 1 + (-0.234 + 0.972i)T \)
19 \( 1 + (-0.266 - 0.963i)T \)
23 \( 1 + (0.834 - 0.551i)T \)
29 \( 1 + (-0.490 + 0.871i)T \)
31 \( 1 + (0.926 + 0.376i)T \)
37 \( 1 + (-0.926 + 0.376i)T \)
41 \( 1 + (0.546 + 0.837i)T \)
43 \( 1 + (0.997 - 0.0660i)T \)
47 \( 1 + (0.287 - 0.957i)T \)
53 \( 1 + (0.982 + 0.186i)T \)
59 \( 1 + (0.528 + 0.849i)T \)
61 \( 1 + (0.381 - 0.924i)T \)
67 \( 1 + (-0.942 + 0.335i)T \)
71 \( 1 + (0.627 - 0.778i)T \)
73 \( 1 + (0.795 + 0.605i)T \)
79 \( 1 + (-0.999 - 0.0110i)T \)
83 \( 1 + (0.894 - 0.446i)T \)
89 \( 1 + (-0.0385 - 0.999i)T \)
97 \( 1 + (-0.828 + 0.560i)T \)
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

−18.112620013262883995097397091369, −17.65621217038073140604015360351, −17.10137256246351823053800930968, −16.31462739773542652324675420097, −15.41123667310630848223616621856, −14.54423408962367068210450277547, −13.74393261640221508768656020875, −13.40807958538377233362499291806, −12.62980775464251974272877464876, −12.11603286178563162002219171611, −11.06549533389800010934671478926, −10.62166545748691955128725490473, −9.8442362986611871654708539825, −9.44577693994586702782797487268, −8.56872381289021845531337589969, −7.8656724068412526819891340345, −7.09460454669556100523623562676, −5.738434421337336871103903677518, −5.4609028293045505549703682138, −4.63559510477478085464912819947, −3.66557076034651308432036171096, −2.576343375640406475180900052813, −2.49967212244769520301587724916, −1.29878244037470596119232635328, −0.40888897539825016407464646508, 1.06380024534191975942299331642, 2.07132313053796324235950591430, 2.82464287981388332804241914069, 4.08306049320607722660553888553, 4.90488288770050413093514205324, 5.3018130327852800974994343203, 6.28021601929254473706873214258, 6.8385563931046892934607408340, 7.387614439462711079463867428699, 8.46488636645637534603539580741, 8.95750695565761270273409693414, 9.661275306656389181356523389188, 10.413876840071909363717584180288, 10.84838857119531262940051991303, 12.19081722005807272641544288880, 12.96194691823451163825351323417, 13.36157285489619266529332417706, 14.14150975468527375276648352750, 14.82289745156138237524032033166, 15.275331680625066662473627120772, 16.146416678485654258514704379748, 16.87360458188307104011515097549, 17.368689648233036773735958289, 17.89521632002056043803085533216, 18.61908223287273055310821644494

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