Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.201 - 0.979i$
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 + (0.396 + 0.918i)3-s + (−0.5 + 0.866i)4-s + (0.286 − 0.957i)5-s + (−0.597 + 0.802i)6-s + (−0.686 − 0.727i)7-s − 8-s + (−0.686 + 0.727i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (−0.993 − 0.116i)12-s + (0.286 − 0.957i)13-s + (0.286 − 0.957i)14-s + (0.993 − 0.116i)15-s + (−0.5 − 0.866i)16-s − 17-s + ⋯
L(s,χ)  = 1  + (0.5 + 0.866i)2-s + (0.396 + 0.918i)3-s + (−0.5 + 0.866i)4-s + (0.286 − 0.957i)5-s + (−0.597 + 0.802i)6-s + (−0.686 − 0.727i)7-s − 8-s + (−0.686 + 0.727i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (−0.993 − 0.116i)12-s + (0.286 − 0.957i)13-s + (0.286 − 0.957i)14-s + (0.993 − 0.116i)15-s + (−0.5 − 0.866i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.201 - 0.979i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.201 - 0.979i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.201 - 0.979i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (21, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.201 - 0.979i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1482933527 + 0.1819077473i$
$L(\frac12,\chi)$  $\approx$  $-0.1482933527 + 0.1819077473i$
$L(\chi,1)$  $\approx$  0.9080185004 + 0.6461532358i
$L(1,\chi)$  $\approx$  0.9080185004 + 0.6461532358i

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

−18.24445104060456104448042608839, −17.73643520267675212661483291397, −16.73766262249210288013035527478, −15.65166116737262772510686289594, −14.8977854792329235119419184757, −14.4270100821279260833299462101, −13.77335405170293164002966685387, −13.20835494368017363372666344759, −12.56718877849467989413807007566, −11.72580341007378087588541992517, −11.383190547735524477918939421, −10.57606304044040278590397104160, −9.59178403489805919431710904897, −8.92288270404689213725782683240, −8.675653015456969868464401355709, −7.051652630781415050359805788859, −6.60310655361272270352626680480, −6.221262309545057576687648321276, −5.277984345314527916985180128914, −4.04831568052747958114109140639, −3.48825356946355851909618046325, −2.52664814367964486755380319375, −2.26509235332485415652809521506, −1.36518567402452229708994774401, −0.0487554532262347067456426999, 1.36333323860946847811057725068, 2.654986048934735060616757305387, 3.614175409616923552588621922534, 4.09712601831215483936554091577, 4.59924545387120911174630556025, 5.7401869091702628662045615982, 5.917146239357975439054187578345, 7.089820865515074762366267537573, 7.815660447007871117975837108089, 8.61722779636107264000935410957, 9.123923465783881300800448318679, 9.77975274740014380894539048945, 10.45601631908123107351264450182, 11.52676822448324106577541967210, 12.34600208648682479727244608425, 13.083642489288824643216674857908, 13.61531288607289307746367628376, 14.15844465470474135827746649385, 15.00835521666374231517042970723, 15.63414634029216263458558604798, 16.18830838645125444363461776044, 16.71291234206160996786430515761, 17.40451947381570594221701928461, 17.72390044499980159278586168609, 19.15963927047537809201701671489

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