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  :  $\chi_{35} (17, \cdot )$
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

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

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