Properties

Degree 1
Conductor $ 5 \cdot 47 $
Sign $0.585 + 0.810i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.942 + 0.334i)2-s + (0.979 + 0.203i)3-s + (0.775 + 0.631i)4-s + (0.854 + 0.519i)6-s + (−0.816 + 0.576i)7-s + (0.519 + 0.854i)8-s + (0.917 + 0.398i)9-s + (−0.460 − 0.887i)11-s + (0.631 + 0.775i)12-s + (0.730 − 0.682i)13-s + (−0.962 + 0.269i)14-s + (0.203 + 0.979i)16-s + (−0.887 − 0.460i)17-s + (0.730 + 0.682i)18-s + (−0.0682 + 0.997i)19-s + ⋯
L(s,χ)  = 1  + (0.942 + 0.334i)2-s + (0.979 + 0.203i)3-s + (0.775 + 0.631i)4-s + (0.854 + 0.519i)6-s + (−0.816 + 0.576i)7-s + (0.519 + 0.854i)8-s + (0.917 + 0.398i)9-s + (−0.460 − 0.887i)11-s + (0.631 + 0.775i)12-s + (0.730 − 0.682i)13-s + (−0.962 + 0.269i)14-s + (0.203 + 0.979i)16-s + (−0.887 − 0.460i)17-s + (0.730 + 0.682i)18-s + (−0.0682 + 0.997i)19-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.585 + 0.810i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.585 + 0.810i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(235\)    =    \(5 \cdot 47\)
\( \varepsilon \)  =  $0.585 + 0.810i$
motivic weight  =  \(0\)
character  :  $\chi_{235} (13, \cdot )$
Sato-Tate  :  $\mu(92)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 235,\ (0:\ ),\ 0.585 + 0.810i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.347094765 + 1.200376410i$
$L(\frac12,\chi)$  $\approx$  $2.347094765 + 1.200376410i$
$L(\chi,1)$  $\approx$  2.054465055 + 0.6949930417i
$L(1,\chi)$  $\approx$  2.054465055 + 0.6949930417i

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

−25.79888802820964286915135730007, −25.379480485070959720894996793162, −23.88301585391734267035136518316, −23.647100562995309459818520295359, −22.32356104681367025282223768516, −21.49546850953119346589159281572, −20.354958828899878207572226533167, −19.95853735934844829549574761620, −19.05302528594293150560168599774, −17.92060714605586223807674651430, −16.222604234647254224373159717, −15.54304220768772396929775680723, −14.56379394624154764241129308020, −13.50049705238271318473119676397, −13.108530349052932086171921464424, −12.03038498910633990193818439950, −10.60643819260535692141197620318, −9.80649407768313255621236209497, −8.55788463682729608648368805110, −7.01511693138517189992672218726, −6.555510672392341981219927554835, −4.70460448155711088509170608546, −3.82629442788147589256899071278, −2.74828601552243083526772841156, −1.61701010852082070341001024307, 2.217941146389984235142626424473, 3.18768375768613863346817335128, 4.023469507990533149504770983462, 5.53574601949013411449788412586, 6.42262492009840514828105519040, 7.84397895887113050058659914656, 8.55784343190196508697798795896, 9.8825256459988701579957191263, 11.093799346068282771141207568294, 12.411697452543163933196441292690, 13.33253467657326088083732625000, 13.875921585319627288522950584560, 15.12395380462230014388089716982, 15.80571149269294525764706238031, 16.40061608845608027493614878132, 18.10845216266609407537101114749, 19.15011055545965423780760289870, 20.13883704357613142335137741146, 20.941688664449014887291067824562, 21.82280190966417775721148398307, 22.58192580147967982959636968051, 23.70029345576023870817000553054, 24.76861423522604479119987201426, 25.2821241326314411772540436289, 26.16919394620397919078376134157

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