Properties

Degree 1
Conductor $ 5 \cdot 7 $
Sign $0.629 + 0.777i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s − 6-s + i·8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)12-s i·13-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.866 − 0.5i)18-s + (−0.5 − 0.866i)19-s i·22-s + (0.866 − 0.5i)23-s + (−0.5 + 0.866i)24-s + ⋯
L(s,χ)  = 1  + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s − 6-s + i·8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)12-s i·13-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.866 − 0.5i)18-s + (−0.5 − 0.866i)19-s i·22-s + (0.866 − 0.5i)23-s + (−0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.629 + 0.777i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.629 + 0.777i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(35\)    =    \(5 \cdot 7\)
\( \varepsilon \)  =  $0.629 + 0.777i$
motivic weight  =  \(0\)
character  :  $\chi_{35} (33, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 35,\ (0:\ ),\ 0.629 + 0.777i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6116804491 + 0.2918893856i$
$L(\frac12,\chi)$  $\approx$  $0.6116804491 + 0.2918893856i$
$L(\chi,1)$  $\approx$  0.7926978904 + 0.2711572794i
$L(1,\chi)$  $\approx$  0.7926978904 + 0.2711572794i

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

−35.96270086253122801640028356519, −35.09296740751776314898562212025, −33.65518575505444040652958940308, −31.78127560345811002225780429535, −30.87003465199653454561197775295, −29.63444560153303177754614178276, −28.732147441629365788061390699132, −27.0177541474684750262038974377, −26.23888240716562225938201244508, −25.057285989709740448874949177, −23.854431123206774184264590205150, −21.62390064472706299153577719837, −20.65134006054562680935884971093, −19.28048453256467808208508314869, −18.63918325735076744651007300612, −17.10377650840336635045748007481, −15.560659752568304742491375763210, −13.776630047602871880605957776918, −12.472599290410091491840568551538, −10.89544887421104472732461505599, −9.22161594293743291918430494614, −8.21038992951406096409383322199, −6.74733222095866922720308575096, −3.57358406320097593820261441512, −1.90826950242874198743818746701, 2.46964557874449569054137389222, 4.93285715404158198128287923772, 7.12060010875330831846743966380, 8.42647097308994696046667306293, 9.66593677165573394486184262328, 10.854025857985630462267460608935, 13.25844629396880483381306945062, 14.977361652386276259721987325088, 15.60366176059770128168371641262, 17.21420766694968521306612889552, 18.566989747968578274225557523680, 19.92702506936952011911507609472, 20.738334712893500248621441580443, 22.63299749653928026841191096058, 24.33717797642689553020890424716, 25.42025122175247016296894445952, 26.313442973567259426520735517494, 27.42257289457287173816836169136, 28.459085286747930218900475285782, 30.10673886355030755446715922793, 31.568146779162031669015226100211, 32.798583434149184894956405391408, 33.66246136474567803781005315002, 35.09475637831373976587767440636, 36.32077338464839187938891941646

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