Properties

Degree 1
Conductor 373
Sign $-0.999 + 0.0288i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.543 − 0.839i)2-s + (−0.0168 − 0.999i)3-s + (−0.409 + 0.912i)4-s + (−0.331 + 0.943i)5-s + (−0.830 + 0.557i)6-s + (−0.820 + 0.571i)7-s + (0.988 − 0.151i)8-s + (−0.999 + 0.0337i)9-s + (0.972 − 0.234i)10-s + (−0.625 + 0.780i)11-s + (0.918 + 0.394i)12-s + (−0.151 + 0.988i)13-s + (0.925 + 0.378i)14-s + (0.948 + 0.315i)15-s + (−0.664 − 0.747i)16-s + (0.994 − 0.101i)17-s + ⋯
L(s,χ)  = 1  + (−0.543 − 0.839i)2-s + (−0.0168 − 0.999i)3-s + (−0.409 + 0.912i)4-s + (−0.331 + 0.943i)5-s + (−0.830 + 0.557i)6-s + (−0.820 + 0.571i)7-s + (0.988 − 0.151i)8-s + (−0.999 + 0.0337i)9-s + (0.972 − 0.234i)10-s + (−0.625 + 0.780i)11-s + (0.918 + 0.394i)12-s + (−0.151 + 0.988i)13-s + (0.925 + 0.378i)14-s + (0.948 + 0.315i)15-s + (−0.664 − 0.747i)16-s + (0.994 − 0.101i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.999 + 0.0288i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.999 + 0.0288i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(373\)
\( \varepsilon \)  =  $-0.999 + 0.0288i$
motivic weight  =  \(0\)
character  :  $\chi_{373} (11, \cdot )$
Sato-Tate  :  $\mu(372)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 373,\ (1:\ ),\ -0.999 + 0.0288i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.001290041494 - 0.08942810690i$
$L(\frac12,\chi)$  $\approx$  $0.001290041494 - 0.08942810690i$
$L(\chi,1)$  $\approx$  0.5261006791 - 0.1533917432i
$L(1,\chi)$  $\approx$  0.5261006791 - 0.1533917432i

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

−25.08915740383369402832731723754, −23.96105169208527029246520321005, −23.10714998199026066350658602231, −22.57809046135844179005521995907, −21.18795476855362788073751811821, −20.36592560562523832210992790473, −19.615842146706713386786505258430, −18.68148193284022242458927377384, −17.35926204687778946981062174962, −16.52597497829141639820705120468, −16.2416061183987557600208279849, −15.39267817181097521218057654361, −14.37589112664490914620531755225, −13.37228523695965923285268006788, −12.3080612733965905286442050450, −10.71334455850696539067267474063, −10.18881696588148959688265182302, −9.23286632332385392397451290384, −8.326914590733616035400545776558, −7.587990485159049811559355606645, −5.95000861060582349576323621548, −5.3775984220084166975060012125, −4.24621304593807277006743099013, −3.17173659809297873152115873398, −0.85283017649068016901803469401, 0.04075334128270818548985181274, 1.67692967206265593937863143658, 2.71595636312831760819349318100, 3.37574872239161128182329746990, 5.11761504988160787838481081539, 6.74552378146524675429931253861, 7.204931201683007307046347915181, 8.30801042189108236068314175721, 9.41504249122550596996105927743, 10.289255520386146664108127127869, 11.52708511511153947521769925675, 11.9929253365900938057076588504, 12.971533164848988905668526159616, 13.77445225396259336973422414580, 14.9407713062316649406808804349, 16.127695497431711807151887700360, 17.23418100049630167529012012388, 18.24300633099665063495401412409, 18.63858726471048859095572316156, 19.43745582980464391409650651842, 20.0166309997814211898237418296, 21.41196410653399088719845271569, 22.12192147409302117913634909321, 23.12205263501107863362584761265, 23.61216782595037546024881547105

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