Properties

Degree 1
Conductor 47
Sign $-0.714 + 0.699i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.917 − 0.398i)2-s + (−0.0682 − 0.997i)3-s + (0.682 + 0.730i)4-s + (−0.854 − 0.519i)5-s + (−0.334 + 0.942i)6-s + (0.203 − 0.979i)7-s + (−0.334 − 0.942i)8-s + (−0.990 + 0.136i)9-s + (0.576 + 0.816i)10-s + (0.775 + 0.631i)11-s + (0.682 − 0.730i)12-s + (−0.962 + 0.269i)13-s + (−0.576 + 0.816i)14-s + (−0.460 + 0.887i)15-s + (−0.0682 + 0.997i)16-s + (−0.775 + 0.631i)17-s + ⋯
L(s,χ)  = 1  + (−0.917 − 0.398i)2-s + (−0.0682 − 0.997i)3-s + (0.682 + 0.730i)4-s + (−0.854 − 0.519i)5-s + (−0.334 + 0.942i)6-s + (0.203 − 0.979i)7-s + (−0.334 − 0.942i)8-s + (−0.990 + 0.136i)9-s + (0.576 + 0.816i)10-s + (0.775 + 0.631i)11-s + (0.682 − 0.730i)12-s + (−0.962 + 0.269i)13-s + (−0.576 + 0.816i)14-s + (−0.460 + 0.887i)15-s + (−0.0682 + 0.997i)16-s + (−0.775 + 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.714 + 0.699i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.714 + 0.699i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(47\)
\( \varepsilon \)  =  $-0.714 + 0.699i$
motivic weight  =  \(0\)
character  :  $\chi_{47} (33, \cdot )$
Sato-Tate  :  $\mu(46)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 47,\ (1:\ ),\ -0.714 + 0.699i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1135762170 - 0.2784196479i$
$L(\frac12,\chi)$  $\approx$  $-0.1135762170 - 0.2784196479i$
$L(\chi,1)$  $\approx$  0.3750170130 - 0.3100761498i
$L(1,\chi)$  $\approx$  0.3750170130 - 0.3100761498i

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

−34.55563525707135773648235966466, −33.72145697057060412101915620842, −32.340388337863069295421813895629, −31.35535790605584471448531385819, −29.609541791101396552945585651138, −28.236768157234460675515563504306, −27.31622417196971824065150305624, −26.78565460871594092958885749815, −25.4240201625381391338837820092, −24.23834112356588598235915945460, −22.68644130443732896714292839601, −21.56990053456855863951463454470, −19.95220945240262342320768147803, −19.08835725879370180134124271025, −17.63375569030629551879442835720, −16.370971803433253671992831015070, −15.25522977071763100184111291268, −14.67375457673813869602505099641, −11.72470790934502343516237946145, −10.94738604246051312937087404502, −9.37556631687425510954884797285, −8.40386204985630538837450864026, −6.67594297506599811061578450793, −5.00349745579655056395374308887, −2.884769344282960895639571203544, 0.23408529944540239693464183976, 1.83046530017871812543218201596, 4.07716730582679223225458845347, 6.85607468781438264787841826540, 7.70975844568035543403873789287, 8.99786490452106930319038362126, 10.84696943758436775126568685493, 12.01656469826791101797862082190, 12.96014312363982413269639158176, 14.89153902374280906263986383116, 16.89364014984045982579229294412, 17.26499213280095852321487998177, 18.99777316365227044177416201607, 19.72475483425169692871112610892, 20.6007206056125267235877382669, 22.62271056474130424263645511911, 23.975964100448772097360346065955, 24.833155766593295006034179380425, 26.27339920995309430123778540346, 27.3833124255925160587092088097, 28.44532380053027562497867762001, 29.61719457211758421475721496741, 30.45971537087582004993291749720, 31.49739823697555441205912720219, 33.398352337115230947074133608444

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