Properties

Degree 1
Conductor $ 2^{7} \cdot 19 $
Sign $0.514 - 0.857i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.980 + 0.195i)3-s + (−0.555 + 0.831i)5-s + (0.923 − 0.382i)7-s + (0.923 + 0.382i)9-s + (−0.195 − 0.980i)11-s + (−0.555 − 0.831i)13-s + (−0.707 + 0.707i)15-s + (−0.707 − 0.707i)17-s + (0.980 − 0.195i)21-s + (−0.382 + 0.923i)23-s + (−0.382 − 0.923i)25-s + (0.831 + 0.555i)27-s + (0.195 − 0.980i)29-s i·31-s i·33-s + ⋯
L(s,χ)  = 1  + (0.980 + 0.195i)3-s + (−0.555 + 0.831i)5-s + (0.923 − 0.382i)7-s + (0.923 + 0.382i)9-s + (−0.195 − 0.980i)11-s + (−0.555 − 0.831i)13-s + (−0.707 + 0.707i)15-s + (−0.707 − 0.707i)17-s + (0.980 − 0.195i)21-s + (−0.382 + 0.923i)23-s + (−0.382 − 0.923i)25-s + (0.831 + 0.555i)27-s + (0.195 − 0.980i)29-s i·31-s i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.514 - 0.857i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.514 - 0.857i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(2432\)    =    \(2^{7} \cdot 19\)
\( \varepsilon \)  =  $0.514 - 0.857i$
motivic weight  =  \(0\)
character  :  $\chi_{2432} (1443, \cdot )$
Sato-Tate  :  $\mu(32)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 2432,\ (0:\ ),\ 0.514 - 0.857i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.720171632 - 0.9744651931i$
$L(\frac12,\chi)$  $\approx$  $1.720171632 - 0.9744651931i$
$L(\chi,1)$  $\approx$  1.362537383 - 0.09260421673i
$L(1,\chi)$  $\approx$  1.362537383 - 0.09260421673i

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

−19.921817381300691969936205411211, −19.0889022582302002407656561717, −18.10576682121359478835790067303, −17.76999610845265948943426304785, −16.635308260641710403217897559680, −16.03670645876865183915429022889, −15.02643548902658328347348406753, −14.81834003023956953329448230980, −14.01892037917800867667660423295, −13.01943609871313022172524120589, −12.48443275387486891714340520950, −11.92235577915657649334695695263, −10.97510756715142965751860779811, −9.98644744525467951169642479491, −9.165567848036633479576255554462, −8.58293208764394355578922125142, −8.025337969732721293619588195904, −7.26674172116628660191243475221, −6.50624053926653836800940073437, −5.096007016969660534538178765217, −4.52899869950760420962997506970, −4.002850173589153325825244800830, −2.726132575307160512319696300273, −1.91314672610273437211409819865, −1.324298090359214342465967181572, 0.53830501121527924791510775519, 1.92882777413069047389627427715, 2.71726679650961385542016182701, 3.40658114216294232753206931413, 4.20921000530118876703603919851, 4.97488508593750769283984708928, 6.06319681975738496972784870407, 7.113877300070051697543850516328, 7.89985470999841229751780351080, 8.012922776993592381221217411943, 9.10704630973650655038463297786, 9.983755984921616325554672184125, 10.657827051055433819296657509439, 11.3562967643412760714996365615, 11.9840564995764741891770410264, 13.40378340684059761401989607679, 13.58783313746996859805553310242, 14.46846612291726864729106989590, 15.10766945618294210644200204367, 15.53681217924398252641938287812, 16.38303721867174326865563527119, 17.38503463122109257310852684901, 18.15549542116375941970038872754, 18.72244158236013561044101414135, 19.50749274480263028564324935307

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