Properties

Degree 1
Conductor 367
Sign $-0.917 + 0.396i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.376 + 0.926i)2-s + (0.952 + 0.304i)3-s + (−0.716 − 0.697i)4-s + (−0.279 + 0.960i)5-s + (−0.640 + 0.767i)6-s + (−0.469 + 0.882i)7-s + (0.916 − 0.400i)8-s + (0.815 + 0.579i)9-s + (−0.784 − 0.620i)10-s + (0.229 − 0.973i)11-s + (−0.469 − 0.882i)12-s + (0.815 + 0.579i)13-s + (−0.640 − 0.767i)14-s + (−0.558 + 0.829i)15-s + (0.0257 + 0.999i)16-s + (0.128 + 0.991i)17-s + ⋯
L(s,χ)  = 1  + (−0.376 + 0.926i)2-s + (0.952 + 0.304i)3-s + (−0.716 − 0.697i)4-s + (−0.279 + 0.960i)5-s + (−0.640 + 0.767i)6-s + (−0.469 + 0.882i)7-s + (0.916 − 0.400i)8-s + (0.815 + 0.579i)9-s + (−0.784 − 0.620i)10-s + (0.229 − 0.973i)11-s + (−0.469 − 0.882i)12-s + (0.815 + 0.579i)13-s + (−0.640 − 0.767i)14-s + (−0.558 + 0.829i)15-s + (0.0257 + 0.999i)16-s + (0.128 + 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.917 + 0.396i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.917 + 0.396i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(367\)
\( \varepsilon \)  =  $-0.917 + 0.396i$
motivic weight  =  \(0\)
character  :  $\chi_{367} (106, \cdot )$
Sato-Tate  :  $\mu(61)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 367,\ (0:\ ),\ -0.917 + 0.396i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2507639949 + 1.212738342i$
$L(\frac12,\chi)$  $\approx$  $0.2507639949 + 1.212738342i$
$L(\chi,1)$  $\approx$  0.7285476954 + 0.7773405557i
$L(1,\chi)$  $\approx$  0.7285476954 + 0.7773405557i

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

−24.33198810575596434550421333162, −23.30580565102900093702680155853, −22.56724025794101822974792361863, −21.121571518133085337680342662511, −20.52414311058344288101198448885, −19.927991443620727629336186969975, −19.45465942168542581350175268148, −18.18463216230120507328833258761, −17.49917428361995881438703566752, −16.34157341347212414299593891227, −15.46232159733152118836821071224, −13.974990508278594313807739886299, −13.29976289289989468778030356714, −12.69096732571402848429626866195, −11.72013968953960023827213297111, −10.475123239600140917750097713697, −9.36484705368578747591984684245, −9.03231542121651950208876462794, −7.68984179188815320749487903519, −7.20464091149144063548457226585, −5.070764361023720772139346007472, −3.95440267398437052548384560242, −3.271446713490825193930352679642, −1.816488807788094086260054325575, −0.8153637920797272986217190685, 1.83172237824296577919260706281, 3.29106019560056878804866494801, 4.055850359639792930076758507530, 5.78108169493404670668685684458, 6.44379207327601467660457642311, 7.65822254716118753013274654623, 8.49503191465190799156608750240, 9.2026599055981437719585708572, 10.23569937352676415478251599567, 11.14424352083230464639923454718, 12.73414847230734938642704608805, 13.86855072167346462173846463361, 14.48128500540958724311319121809, 15.26413578254268941742955367949, 15.993636390997836220597593562284, 16.767819022165089790176437242842, 18.39723956619518808444906156986, 18.835198364699798445138708406095, 19.256046849776377780068162069890, 20.60730917098677411910726246975, 21.882992144576514473864809638194, 22.28279797189394735937852068786, 23.48921904603210840090466049240, 24.43984179708087719284281475780, 25.18580355472731177261749324904

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