Properties

Degree 1
Conductor 373
Sign $0.528 - 0.849i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.612 − 0.790i)2-s + (−0.954 + 0.299i)3-s + (−0.250 − 0.968i)4-s + (−0.979 + 0.201i)5-s + (−0.347 + 0.937i)6-s + (−0.0506 + 0.998i)7-s + (−0.918 − 0.394i)8-s + (0.820 − 0.571i)9-s + (−0.440 + 0.897i)10-s + (−0.918 + 0.394i)11-s + (0.528 + 0.848i)12-s + (0.918 − 0.394i)13-s + (0.758 + 0.651i)14-s + (0.874 − 0.485i)15-s + (−0.874 + 0.485i)16-s + (−0.250 − 0.968i)17-s + ⋯
L(s,χ)  = 1  + (0.612 − 0.790i)2-s + (−0.954 + 0.299i)3-s + (−0.250 − 0.968i)4-s + (−0.979 + 0.201i)5-s + (−0.347 + 0.937i)6-s + (−0.0506 + 0.998i)7-s + (−0.918 − 0.394i)8-s + (0.820 − 0.571i)9-s + (−0.440 + 0.897i)10-s + (−0.918 + 0.394i)11-s + (0.528 + 0.848i)12-s + (0.918 − 0.394i)13-s + (0.758 + 0.651i)14-s + (0.874 − 0.485i)15-s + (−0.874 + 0.485i)16-s + (−0.250 − 0.968i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.528 - 0.849i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.528 - 0.849i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(373\)
\( \varepsilon \)  =  $0.528 - 0.849i$
motivic weight  =  \(0\)
character  :  $\chi_{373} (31, \cdot )$
Sato-Tate  :  $\mu(62)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 373,\ (0:\ ),\ 0.528 - 0.849i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8430503884 - 0.4684239780i$
$L(\frac12,\chi)$  $\approx$  $0.8430503884 - 0.4684239780i$
$L(\chi,1)$  $\approx$  0.8459338297 - 0.2959251574i
$L(1,\chi)$  $\approx$  0.8459338297 - 0.2959251574i

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

−24.23261868740860701840232074542, −23.731034965041594304572348141858, −23.317647709127225712581354695388, −22.52882478883882353116893648267, −21.47539012114766039017472499875, −20.600139791632017695827074593074, −19.39162086289375492172489435696, −18.333796504470792802259877333669, −17.50495592921225515909532180515, −16.42439614850966713262068696966, −16.14445048184954637333719618780, −15.1995129745089192226471868043, −13.85312958448122026792608673541, −13.09241186414733434682925306259, −12.37192405187014252730928676078, −11.19702340086974214735958815658, −10.69597853582673691285922794625, −8.82584046052596107750276769994, −7.77969989434363638339372816623, −7.12059178494842137683789038726, −6.188582523593227191796987201181, −5.02606058061684623766348042455, −4.26401327128057484386726882413, −3.21732608107502594957027424965, −0.92732552028908384552752973407, 0.78637674227435147831721047876, 2.56512172649110956039339013581, 3.5899776131580067490521653454, 4.7486809297423836520385651082, 5.46204554499623448865443210441, 6.4729731469366723399223217117, 7.833306603207703304025162271864, 9.24979231749213048134515045588, 10.18957615288084613371257812188, 11.27952009492520583364694264750, 11.63097601838623691080805076947, 12.55336114198034611924527559331, 13.3718128366032620059160043211, 14.882946016388898015018824965338, 15.64768176267480472290208374723, 15.98346826447436956542645226400, 17.74301827711324628255178498219, 18.546418015514451949084468035530, 18.98311474237457021245061475439, 20.516014440108190919356972899733, 20.88304193222675084499785558305, 22.05446288060600909304020498219, 22.78527044634875798848388378698, 23.17151043068616620844501358021, 24.1098598128030350102435118269

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