Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.247 - 0.968i$
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 + (0.993 + 0.116i)3-s + (0.5 − 0.866i)4-s + (−0.727 + 0.686i)5-s + (0.918 − 0.396i)6-s + (0.973 − 0.230i)7-s i·8-s + (0.973 + 0.230i)9-s + (−0.286 + 0.957i)10-s + (0.286 + 0.957i)11-s + (0.597 − 0.802i)12-s + (−0.727 + 0.686i)13-s + (0.727 − 0.686i)14-s + (−0.802 + 0.597i)15-s + (−0.5 − 0.866i)16-s i·17-s + ⋯
L(s,χ)  = 1  + (0.866 − 0.5i)2-s + (0.993 + 0.116i)3-s + (0.5 − 0.866i)4-s + (−0.727 + 0.686i)5-s + (0.918 − 0.396i)6-s + (0.973 − 0.230i)7-s i·8-s + (0.973 + 0.230i)9-s + (−0.286 + 0.957i)10-s + (0.286 + 0.957i)11-s + (0.597 − 0.802i)12-s + (−0.727 + 0.686i)13-s + (0.727 − 0.686i)14-s + (−0.802 + 0.597i)15-s + (−0.5 − 0.866i)16-s i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.247 - 0.968i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.247 - 0.968i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.247 - 0.968i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (15, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (1:\ ),\ -0.247 - 0.968i)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.192537708 - 4.111724455i$
$L(\frac12,\chi)$  $\approx$  $3.192537708 - 4.111724455i$
$L(\chi,1)$  $\approx$  2.170602732 - 0.7080777549i
$L(1,\chi)$  $\approx$  2.170602732 - 0.7080777549i

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

−18.597861677143831749637162191849, −17.51612467311038230074963778480, −17.15117540723230461905283371970, −16.094731584474443695871823376338, −15.6924705441911753735331266321, −14.94611833238864746836914995468, −14.46354812580429560329873493558, −13.91536888545796567571332084830, −13.06243669361250811004312210236, −12.36492010454059494397185275869, −12.09155638556305876311346135178, −11.03096007154729337894975075243, −10.37707904072282224284747924935, −8.99397260738480780515046361401, −8.48105570298494394845725408620, −8.05175415585455526606164393619, −7.53524499436693471190581116696, −6.54045996246243620462035823987, −5.686690872738560517327493276904, −4.8634847791783149521190851547, −4.20845223306842331413711462336, −3.63770898938044600285650435339, −2.78775577581301948511978620115, −1.928672501609087864182300627805, −1.042192762437654412066465208619, 0.432888578014839274287781840512, 1.70955539216852449250332155418, 2.29256096035655878133786849303, 2.85388787306883635168554201692, 3.93997733156457116361441124896, 4.53442864866455320054772611883, 4.66331286349815889256206299183, 6.20273087084930487538297732879, 6.97702880934434609589635143843, 7.51310414626373994997306294681, 8.17833854380217147980765250246, 9.2505619381415536898865453094, 9.980362797649610431597095632521, 10.482686264897592343395517371919, 11.51088067151875041914710993757, 11.81438689506357001729284827907, 12.60340425591857279524504971759, 13.5083476714362561739162234568, 14.22790812991722754070037197312, 14.544549582426939567539072226881, 15.09574804234929327224223356890, 15.66938469807763443723845105094, 16.47514809835692966270117992924, 17.62201696678110947571856766016, 18.34946454024975870047394634539

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