↑

Properties

Label 1-6041-6041.454-r0-0-0
Degree $1$
Conductor $6041$
Sign $-0.645 - 0.764i$
Analytic cond. $28.0542$
Root an. cond. $28.0542$
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  + (0.227 + 0.973i)2-s + (0.373 − 0.927i)3-s + (−0.896 + 0.443i)4-s + (0.629 − 0.776i)5-s + (0.988 + 0.152i)6-s + (−0.635 − 0.771i)8-s + (−0.721 − 0.692i)9-s + (0.899 + 0.436i)10-s + (0.565 − 0.824i)11-s + (0.0764 + 0.997i)12-s + (0.997 − 0.0728i)13-s + (−0.485 − 0.874i)15-s + (0.607 − 0.794i)16-s + (−0.934 − 0.356i)17-s + (0.510 − 0.859i)18-s + (−0.535 − 0.844i)19-s + ⋯
L(s)  = 1  + (0.227 + 0.973i)2-s + (0.373 − 0.927i)3-s + (−0.896 + 0.443i)4-s + (0.629 − 0.776i)5-s + (0.988 + 0.152i)6-s + (−0.635 − 0.771i)8-s + (−0.721 − 0.692i)9-s + (0.899 + 0.436i)10-s + (0.565 − 0.824i)11-s + (0.0764 + 0.997i)12-s + (0.997 − 0.0728i)13-s + (−0.485 − 0.874i)15-s + (0.607 − 0.794i)16-s + (−0.934 − 0.356i)17-s + (0.510 − 0.859i)18-s + (−0.535 − 0.844i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.645 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.645 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6041\)    =    \(7 \cdot 863\)
Sign: $-0.645 - 0.764i$
Analytic conductor: \(28.0542\)
Root analytic conductor: \(28.0542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6041} (454, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6041,\ (0:\ ),\ -0.645 - 0.764i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7387197413 - 1.590393461i\)
\(L(\frac12)\) \(\approx\) \(0.7387197413 - 1.590393461i\)
\(L(1)\) \(\approx\) \(1.220969847 - 0.2605067574i\)
\(L(1)\) \(\approx\) \(1.220969847 - 0.2605067574i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
863 \( 1 \)
good2 \( 1 + (0.227 + 0.973i)T \)
3 \( 1 + (0.373 - 0.927i)T \)
5 \( 1 + (0.629 - 0.776i)T \)
11 \( 1 + (0.565 - 0.824i)T \)
13 \( 1 + (0.997 - 0.0728i)T \)
17 \( 1 + (-0.934 - 0.356i)T \)
19 \( 1 + (-0.535 - 0.844i)T \)
23 \( 1 + (-0.984 + 0.174i)T \)
29 \( 1 + (0.184 - 0.982i)T \)
31 \( 1 + (0.119 - 0.992i)T \)
37 \( 1 + (0.0982 - 0.995i)T \)
41 \( 1 + (0.0619 + 0.998i)T \)
43 \( 1 + (0.987 + 0.159i)T \)
47 \( 1 + (0.170 + 0.985i)T \)
53 \( 1 + (-0.400 + 0.916i)T \)
59 \( 1 + (0.769 - 0.638i)T \)
61 \( 1 + (-0.0255 - 0.999i)T \)
67 \( 1 + (0.577 + 0.816i)T \)
71 \( 1 + (-0.778 - 0.627i)T \)
73 \( 1 + (-0.00364 + 0.999i)T \)
79 \( 1 + (-0.433 - 0.901i)T \)
83 \( 1 + (-0.861 - 0.507i)T \)
89 \( 1 + (0.641 - 0.767i)T \)
97 \( 1 + (-0.951 + 0.308i)T \)
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

−17.91553927391291009026249232651, −17.55852606867584654481670210566, −16.68149608582290251087373935673, −15.81899394148286915294820826329, −15.04870819278901871640387094802, −14.61982897910203931875117476250, −13.94850939014752896858046505194, −13.55775283625433643956445051849, −12.63909752709664997424717687058, −11.9350386858840108627775143792, −11.0743567403472826122581600939, −10.63440652670367819149456164452, −10.1025049243072812642834270630, −9.57411861229312439020648527154, −8.67867042432263512460189730775, −8.44039490589987073242014477296, −7.08103492211844067507338789977, −6.25129372767582104546808166777, −5.63685134131368936668715182302, −4.77292656230428256733796649722, −3.95976201242039717310948114124, −3.66066730069490142116133242281, −2.71662238856461778269670630440, −2.02546628975780936150573022860, −1.459034420828933542162750436878, 0.391567442332693044713617471885, 1.07460865743927289545134157901, 2.12176514923434421510837969400, 2.91189438987865684686149072318, 3.99882758222264271057984752649, 4.43860020587218353677794529645, 5.604128723374256619177316045378, 6.18593279441017349348819554399, 6.36592189275897080543157612558, 7.4134881586680448657550716202, 8.13146015676988812843478293962, 8.65108706187561682981933727603, 9.15690497123183309890696031883, 9.72552576764805166486819923735, 11.09277596669933879950668847587, 11.6575226033276268219768197639, 12.60873264657492593470284113199, 13.09495388381128505375068872247, 13.608903598182277636399620065939, 14.00941865427740227632648417900, 14.68564360967896832104025769706, 15.718189366354820039744048763748, 16.01417516258066868509128507664, 16.92521831811348769159568107677, 17.56799580656778408374836267275

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