Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.872 + 0.488i$
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 + (0.597 + 0.802i)3-s + (−0.5 − 0.866i)4-s + (−0.973 − 0.230i)5-s + (0.993 − 0.116i)6-s + (−0.286 − 0.957i)7-s − 8-s + (−0.286 + 0.957i)9-s + (−0.686 + 0.727i)10-s + (−0.686 − 0.727i)11-s + (0.396 − 0.918i)12-s + (−0.973 − 0.230i)13-s + (−0.973 − 0.230i)14-s + (−0.396 − 0.918i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯
L(s,χ)  = 1  + (0.5 − 0.866i)2-s + (0.597 + 0.802i)3-s + (−0.5 − 0.866i)4-s + (−0.973 − 0.230i)5-s + (0.993 − 0.116i)6-s + (−0.286 − 0.957i)7-s − 8-s + (−0.286 + 0.957i)9-s + (−0.686 + 0.727i)10-s + (−0.686 − 0.727i)11-s + (0.396 − 0.918i)12-s + (−0.973 − 0.230i)13-s + (−0.973 − 0.230i)14-s + (−0.396 − 0.918i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.872 + 0.488i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.872 + 0.488i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.872 + 0.488i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (548, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.872 + 0.488i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3124812424 + 0.08161110311i$
$L(\frac12,\chi)$  $\approx$  $0.3124812424 + 0.08161110311i$
$L(\chi,1)$  $\approx$  0.7584272358 - 0.4107642655i
$L(1,\chi)$  $\approx$  0.7584272358 - 0.4107642655i

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

−18.39436880746482300073087801083, −17.82905498454134935196365994857, −17.04392263605359825947456045255, −16.11777863719181416557445032867, −15.48918400391912299471094707928, −14.85062588345023733839302751122, −14.74736752708633775513790341699, −13.58773603706401398065447530456, −12.995322391780801332520380326464, −12.32904956280234490401573392330, −11.98948580200933936977368406939, −11.171281516908332244094374544433, −9.70588633902425078831502138848, −9.25463147600521279874333682195, −8.23094723333210065927227255141, −7.94654340957280687712274644194, −7.15871538382796905473847480875, −6.70420533718828889028681761936, −5.77598502425212442823884144103, −5.04144001880222563430892469498, −4.08096245173507909282801350534, −3.43687257822161382228945240957, −2.531003212138998717186325365909, −2.00139153755429092014028378504, −0.10163772898283026832400883124, 0.6415939672606209098753539293, 2.10291779887210455286140437939, 2.88713604499055916434374241992, 3.43909819084501634372117545468, 4.24646030911360269996668819236, 4.703254702015591154274903973953, 5.32542642404981829623312978900, 6.60155992322800848592236340219, 7.42269662133497033137472273528, 8.318031897205525111954706108515, 8.87604830388005676460283178954, 9.68082471381415362923765359029, 10.45982123625203276568439938909, 10.9049245256490248862275006032, 11.4311046837524379626307863041, 12.46865061588586371615628966863, 13.11500108610638152615617526580, 13.64831733422278056379789491584, 14.40238052853561151733205588975, 15.104745174638069222284660020571, 15.61720245653272347270814089166, 16.3247254689933204702463425690, 17.016942853320104931989570632401, 17.98661745343852713500800104698, 18.96465562732528797232481458140

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