Properties

Degree 1
Conductor $ 3 \cdot 7 \cdot 41 $
Sign $0.331 - 0.943i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s i·11-s i·13-s + 16-s i·17-s + i·19-s − 20-s i·22-s − 23-s + 25-s i·26-s + ⋯
L(s,χ)  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s i·11-s i·13-s + 16-s i·17-s + i·19-s − 20-s i·22-s − 23-s + 25-s i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.331 - 0.943i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 861 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.331 - 0.943i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(861\)    =    \(3 \cdot 7 \cdot 41\)
\( \varepsilon \)  =  $0.331 - 0.943i$
motivic weight  =  \(0\)
character  :  $\chi_{861} (419, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 861,\ (0:\ ),\ 0.331 - 0.943i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.767479367 - 1.253069035i$
$L(\frac12,\chi)$  $\approx$  $1.767479367 - 1.253069035i$
$L(\chi,1)$  $\approx$  1.589603913 - 0.3530832894i
$L(1,\chi)$  $\approx$  1.589603913 - 0.3530832894i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−22.088377403407637040337317756, −21.74791515121004186155477421367, −20.55341613031707766751288542852, −19.95769758982572868291214449967, −19.39342972389431100406751991456, −18.35799038985356725036151239434, −17.224665158360562079059097783733, −16.31599909885051674779710705201, −15.71875690829247075560815685366, −14.816938527511606914166318705727, −14.416161397044960704086058409338, −13.15848569817929130285177149850, −12.55024364846094037446474634327, −11.74771421700038666166194799771, −11.12023299841340358754539448344, −10.1743948739919451637917823574, −8.96484689154316333199253044026, −7.859992898865770273684043199831, −7.10131874543822702866046801746, −6.40284330306643662542458287277, −5.14503527141934851451972699948, −4.303218107160619619715549924880, −3.764143959160724146519671177452, −2.53714251435897216724625098013, −1.52309403706327487797313311379, 0.69189950075183436600003873732, 2.27207211611163152896332612142, 3.3866891889732090274243651703, 3.84364978847679906119214957593, 5.039492171567203643807727328605, 5.786826812723731445502802972520, 6.7727013864339777859153446620, 7.85209515860951260070500157974, 8.229134817180741600796464165094, 9.79987262447290047202349062822, 10.77596173387016866979262019702, 11.5296419863353649428178204918, 12.118323946887964962822706310639, 13.03643841071074552112060305180, 13.81585442644982229225949198614, 14.700035047538261250303728397184, 15.37299043274541641892958509126, 16.27991323172266913309894053782, 16.56923791516984112990892732272, 18.06819765023634595154082144080, 18.89596665580660832032431817895, 19.80229882003119665892658982930, 20.33578837809524063920260658829, 21.134607012142341687646242570481, 22.13223098770157619715024404504

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