Properties

Degree 1
Conductor 37
Sign $0.0789 - 0.996i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.984 − 0.173i)2-s + (−0.173 − 0.984i)3-s + (0.939 + 0.342i)4-s + (0.642 + 0.766i)5-s + i·6-s + (0.766 − 0.642i)7-s + (−0.866 − 0.5i)8-s + (−0.939 + 0.342i)9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.173 − 0.984i)12-s + (0.342 − 0.939i)13-s + (−0.866 + 0.5i)14-s + (0.642 − 0.766i)15-s + (0.766 + 0.642i)16-s + (−0.342 − 0.939i)17-s + ⋯
L(s,χ)  = 1  + (−0.984 − 0.173i)2-s + (−0.173 − 0.984i)3-s + (0.939 + 0.342i)4-s + (0.642 + 0.766i)5-s + i·6-s + (0.766 − 0.642i)7-s + (−0.866 − 0.5i)8-s + (−0.939 + 0.342i)9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.173 − 0.984i)12-s + (0.342 − 0.939i)13-s + (−0.866 + 0.5i)14-s + (0.642 − 0.766i)15-s + (0.766 + 0.642i)16-s + (−0.342 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.0789 - 0.996i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.0789 - 0.996i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(37\)
\( \varepsilon \)  =  $0.0789 - 0.996i$
motivic weight  =  \(0\)
character  :  $\chi_{37} (35, \cdot )$
Sato-Tate  :  $\mu(36)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 37,\ (1:\ ),\ 0.0789 - 0.996i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7424946621 - 0.6859944659i$
$L(\frac12,\chi)$  $\approx$  $0.7424946621 - 0.6859944659i$
$L(\chi,1)$  $\approx$  0.7405587630 - 0.3448346246i
$L(1,\chi)$  $\approx$  0.7405587630 - 0.3448346246i

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

−35.69265126019711906548978380233, −34.25443625293734794547290623033, −33.3913210225004832575332392516, −32.49433029481102895109984289114, −30.812522218149469970558660232618, −28.93382613784864913166591328158, −28.18311552519937553454207981322, −27.405213041633679852039161724403, −25.94839024926487264403537693880, −25.03807341961331090152119562507, −23.65000435751342850112201361444, −21.52090832484903935324943719480, −20.92961483285619513912190427817, −19.53425270623199901434158644371, −17.66702545048274254585487788417, −17.04086197485321908831969264644, −15.65999750188318630994890376847, −14.54182891544918148362582291803, −12.10267671937071235581071052167, −10.74452865777710369110020323070, −9.31668898970906494136369960411, −8.58851187424847625274515470083, −6.23900896276632979789321686659, −4.70551158111097392973142722457, −1.85723336067379756756939573408, 1.01296372560939616750146823152, 2.72293432365687372094471494947, 6.111995538910197666720272470838, 7.29727657323673763886901764165, 8.6179106495501684897860326766, 10.58344638492487942410959519731, 11.42959955553886523970340098990, 13.26215232727203559158126451401, 14.64494574194757745044474147771, 16.78215966523864947067816517442, 17.764309387085154935933397143505, 18.58088252642171348447316172644, 19.83746308673786457707800612300, 21.21396058961415983459070573831, 22.88228774597401294128126631998, 24.47274016937534036874840622607, 25.29224899683971509032954304917, 26.61125659354295907921302928800, 27.737701693006436982990663555286, 29.4423158124005981118435442874, 29.7645655461738433195241518829, 30.83434067524772617003313019800, 33.12627368920206119726593397671, 34.23629922325635716340407847932, 35.03485256578787262756060956440

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