Properties

Degree 1
Conductor $ 7 \cdot 379 $
Sign $-0.895 + 0.444i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 6-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s − 13-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s,χ)  = 1  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 6-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s − 13-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 2653 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.895 + 0.444i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 2653 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.895 + 0.444i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(2653\)    =    \(7 \cdot 379\)
\( \varepsilon \)  =  $-0.895 + 0.444i$
motivic weight  =  \(0\)
character  :  $\chi_{2653} (1136, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 2653,\ (1:\ ),\ -0.895 + 0.444i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.2894365087 - 1.235528582i$
$L(\frac12,\chi)$  $\approx$  $-0.2894365087 - 1.235528582i$
$L(\chi,1)$  $\approx$  1.077489175 - 0.2983175472i
$L(1,\chi)$  $\approx$  1.077489175 - 0.2983175472i

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

−19.74884619532890708818012555596, −19.16185314172346021332314315572, −18.44348018769374877695337104997, −17.45665140900127933387334217689, −16.76123225999776325423995133068, −15.612109304568464272452673011884, −14.98839407284286690121478459384, −14.687625404646810107441425778120, −14.0653149238610053952009314110, −13.173767647540448710593614392977, −12.05058253561567895580370477295, −11.88912519793084273617401268824, −10.7378164113370750125958147141, −10.2378689658682696250400396055, −9.802194448791549531025317663579, −8.91362080514195446361016952228, −7.98133443078392432636703158692, −7.176616944852387979032039293806, −6.097006350360254706685558354784, −5.27151461796858486417537627055, −4.3170670644905237640216047379, −3.856465346695122580305326491120, −3.15107891524638554260006909583, −2.29306026091443629352588316158, −1.56288172012744771420297650147, 0.22420386904056536672222333324, 0.63962837752425527590496931811, 2.11184196799349573426181428471, 3.005562367302350729297128176568, 3.859142015362288827088251973698, 4.64691779428515980575511602965, 5.49989772996511209497782490589, 6.26829933081384671801192448718, 7.12137852011322353890180933974, 7.70239063944592213837667438398, 8.3493876698564513606807396986, 9.05990730300263621321045217582, 9.55437394622400320025300266325, 11.232824144474331345641991931904, 11.94089543215699220083335194151, 12.43372784218633542250419815679, 13.17401245282227175549899667558, 13.72021312221937215031675364106, 14.54591703179930507714266806272, 14.979973422299770348028719392621, 16.00733891288893964728043590967, 16.541694165066242468292976996880, 17.24998631987843060397234555877, 17.830532753953775718341274492517, 18.91655283484630373293530701655

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