Properties

Degree 1
Conductor 29
Sign $0.855 + 0.517i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.623 + 0.781i)2-s + (−0.222 − 0.974i)3-s + (−0.222 + 0.974i)4-s + (0.623 + 0.781i)5-s + (0.623 − 0.781i)6-s + (−0.222 − 0.974i)7-s + (−0.900 + 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.222 + 0.974i)10-s + (−0.900 − 0.433i)11-s + 12-s + (−0.900 − 0.433i)13-s + (0.623 − 0.781i)14-s + (0.623 − 0.781i)15-s + (−0.900 − 0.433i)16-s + 17-s + ⋯
L(s,χ)  = 1  + (0.623 + 0.781i)2-s + (−0.222 − 0.974i)3-s + (−0.222 + 0.974i)4-s + (0.623 + 0.781i)5-s + (0.623 − 0.781i)6-s + (−0.222 − 0.974i)7-s + (−0.900 + 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.222 + 0.974i)10-s + (−0.900 − 0.433i)11-s + 12-s + (−0.900 − 0.433i)13-s + (0.623 − 0.781i)14-s + (0.623 − 0.781i)15-s + (−0.900 − 0.433i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.855 + 0.517i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.855 + 0.517i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.855 + 0.517i$
motivic weight  =  \(0\)
character  :  $\chi_{29} (16, \cdot )$
Sato-Tate  :  $\mu(7)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 29,\ (0:\ ),\ 0.855 + 0.517i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8670025821 + 0.2419789471i$
$L(\frac12,\chi)$  $\approx$  $0.8670025821 + 0.2419789471i$
$L(\chi,1)$  $\approx$  1.107935304 + 0.2465575357i
$L(1,\chi)$  $\approx$  1.107935304 + 0.2465575357i

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

−37.39027899654660773251304173249, −36.49248713724828751864705575186, −34.32700190206809129728835216859, −33.17066664678966439088990420063, −32.00319791325642011574843912590, −31.43322853231579063423886782330, −29.38177252567326692571952768931, −28.46364549863046286890333186446, −27.71427759350211912667125558644, −25.83854410020353207462493594906, −24.25221451545722138060684098177, −22.75725135996340718917740299621, −21.463178604150063307236876344128, −20.99954262032163585143748860372, −19.42733449376387309090769850350, −17.62570589298972592738392630703, −15.91086718365607048026202776731, −14.69076706307787309494151081602, −12.96277523568800026875007687731, −11.74763329073383371968176783727, −10.02645301479179505306856722472, −9.14032345328068345315784728183, −5.66282989807631649012267817691, −4.76049809736198473232714243615, −2.61651033357808942884242111086, 2.95339600789876898960738874757, 5.49183744252799304859137727700, 6.84017801857655669265959072418, 7.89168470411729630356460724305, 10.478710533679796669406020076019, 12.449968794676153735961084861045, 13.63032461782041315813390405205, 14.56242637685005727140846513990, 16.617693794563357237382129598645, 17.643664362639249722445539823171, 18.93799746367965548045941164598, 20.959154739538277713774750518522, 22.58576655421226264164138668789, 23.33703394391592055027966476857, 24.654817954722246583627492623241, 25.75999515109237803186809704530, 26.83677781008153409066475554693, 29.34434995150138709418098390670, 29.87422545017352444003071125672, 31.15216951447500548655986026265, 32.565642483692090955492166232507, 33.97439334764516360642997561299, 34.54052223603958905751855953794, 36.07784148589745953785555874544, 36.94961694925969965086927833969

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