Properties

Degree 1
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $0.996 - 0.0893i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.280 + 0.959i)2-s + (−0.842 + 0.538i)4-s + (−0.599 − 0.800i)5-s + (−0.753 − 0.657i)8-s + (0.599 − 0.800i)10-s + (0.646 + 0.762i)11-s + (0.691 − 0.722i)13-s + (0.420 − 0.907i)16-s + (0.887 − 0.460i)17-s + (0.978 + 0.207i)19-s + (0.936 + 0.351i)20-s + (−0.550 + 0.834i)22-s + (0.163 − 0.986i)23-s + (−0.280 + 0.959i)25-s + (0.887 + 0.460i)26-s + ⋯
L(s,χ)  = 1  + (0.280 + 0.959i)2-s + (−0.842 + 0.538i)4-s + (−0.599 − 0.800i)5-s + (−0.753 − 0.657i)8-s + (0.599 − 0.800i)10-s + (0.646 + 0.762i)11-s + (0.691 − 0.722i)13-s + (0.420 − 0.907i)16-s + (0.887 − 0.460i)17-s + (0.978 + 0.207i)19-s + (0.936 + 0.351i)20-s + (−0.550 + 0.834i)22-s + (0.163 − 0.986i)23-s + (−0.280 + 0.959i)25-s + (0.887 + 0.460i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.996 - 0.0893i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.996 - 0.0893i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $0.996 - 0.0893i$
motivic weight  =  \(0\)
character  :  $\chi_{6027} (857, \cdot )$
Sato-Tate  :  $\mu(210)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6027,\ (0:\ ),\ 0.996 - 0.0893i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.863074137 - 0.08339111316i$
$L(\frac12,\chi)$  $\approx$  $1.863074137 - 0.08339111316i$
$L(\chi,1)$  $\approx$  1.112534733 + 0.2887668065i
$L(1,\chi)$  $\approx$  1.112534733 + 0.2887668065i

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

−17.97184943656452857884331619547, −17.21980120864452829782929334022, −16.29605097013423776270294102088, −15.73797914145260768015658857751, −14.8454095720301144703033899760, −14.29614901975242263020103674941, −13.818177940715817598805512453748, −13.194085897387505485748217442782, −12.04502703985043913209655146640, −11.88528002582369284774054287740, −11.13108756375196484105925185947, −10.674605907695891684939362129648, −9.88652005590561505285807435903, −9.15239624231252121110889277772, −8.54165946335109227818520826477, −7.71881403293187249840244508802, −6.89816071059114739682724082381, −6.07019867984285432684758915853, −5.51130860200113396562623014885, −4.47403806231846183218729764484, −3.751982284730413789752158828751, −3.3160678952441699902955867983, −2.67965938676239688688208434417, −1.480485032737582375964862946709, −0.993326835346838331122157384787, 0.56527982600165689808374503505, 1.225336884920487758782812870892, 2.61495538384689773690571627516, 3.66155156936025684386046398861, 3.98111885310939133325507280805, 4.95338280674974656997873145657, 5.37221233230224017534514139730, 6.16902090110130974008467214658, 7.04195524599406911060382972896, 7.608988145084951586560840985081, 8.22091977399565415471796275619, 8.85124867347297212031907849605, 9.54234927631383216241653119457, 10.15690105371601486914804502134, 11.343847856251589922072723043086, 11.96887035224502626613409247802, 12.641079639609742842494669335912, 12.98586273019494388777401877704, 13.94710303651399217940045187531, 14.509232658922917906468110838781, 15.10721556782061070954356215549, 15.979187847925995114261067671173, 16.158153815877069122685747575205, 16.8948869891490923251552048167, 17.564759723160515152262275145664

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