Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.890 - 0.455i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.173 − 0.984i)2-s + (0.396 + 0.918i)3-s + (−0.939 − 0.342i)4-s + (−0.686 − 0.727i)5-s + (0.973 − 0.230i)6-s + (0.973 − 0.230i)7-s + (−0.5 + 0.866i)8-s + (−0.686 + 0.727i)9-s + (−0.835 + 0.549i)10-s + (−0.835 − 0.549i)11-s + (−0.0581 − 0.998i)12-s + (−0.286 + 0.957i)13-s + (−0.0581 − 0.998i)14-s + (0.396 − 0.918i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯
L(s,χ)  = 1  + (0.173 − 0.984i)2-s + (0.396 + 0.918i)3-s + (−0.939 − 0.342i)4-s + (−0.686 − 0.727i)5-s + (0.973 − 0.230i)6-s + (0.973 − 0.230i)7-s + (−0.5 + 0.866i)8-s + (−0.686 + 0.727i)9-s + (−0.835 + 0.549i)10-s + (−0.835 − 0.549i)11-s + (−0.0581 − 0.998i)12-s + (−0.286 + 0.957i)13-s + (−0.0581 − 0.998i)14-s + (0.396 − 0.918i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.890 - 0.455i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.890 - 0.455i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.890 - 0.455i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (118, \cdot )$
Sato-Tate  :  $\mu(27)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.890 - 0.455i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1837513453 - 0.7631348154i$
$L(\frac12,\chi)$  $\approx$  $0.1837513453 - 0.7631348154i$
$L(\chi,1)$  $\approx$  0.8593525365 - 0.3330260107i
$L(1,\chi)$  $\approx$  0.8593525365 - 0.3330260107i

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

−18.426596520376918881687668470154, −18.19879331796327087745717802568, −17.39362742037038297579030915505, −17.002661075116574163718719142031, −15.57362523688352986283873571428, −15.33649511768199924371402866417, −14.62799966432115443748455389990, −14.37052183416860151652648799363, −13.2697283135839177738043202931, −12.70603089456523331334874318043, −12.297066194832186351979957813592, −11.120647471450660641094048330464, −10.685977447377547425400304312326, −9.49127602267053905424285357241, −8.619681741997917414166427553328, −8.04003638165059054561381773581, −7.53866257358892404916634369584, −7.136392325987804248744548596767, −6.15496095396502350892690901089, −5.52889091107997373769507223808, −4.62420538039653640502337954070, −3.85523308642816343078258513188, −2.83891817118463030268992099195, −2.28971304039336422909278887942, −0.94180469491077196949181158076, 0.22808214711911979871605507106, 1.474540209928333470673220107355, 2.21793550030832980164987455363, 3.23148828592542515024226439526, 3.870401278980880091367810663225, 4.49409466648898815374561403429, 5.22807069038615519525579854016, 5.496848953527053462318111594919, 7.29653673080147348687222315925, 7.99752633774089161994672440978, 8.64640151646752248236735131980, 9.20350700094420818871481041002, 9.87706812113120190830032057504, 10.851109699987525692319037951846, 11.21969772042947512862522123253, 11.75119719350003708764577180914, 12.67456054437711410273112866126, 13.38009042879128377090733464249, 14.14754478349969206644677557271, 14.61773558561813177308158824876, 15.32930677684092425253993125146, 16.25468108406382517888153749196, 16.69787713795539758906754419632, 17.45945432468282593445548162654, 18.530051260386591637744401314771

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