Properties

Degree 1
Conductor $ 5 \cdot 17 $
Sign $0.564 - 0.825i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s + (0.707 − 0.707i)3-s + 4-s + (0.707 − 0.707i)6-s + (−0.707 − 0.707i)7-s + 8-s i·9-s + (0.707 − 0.707i)11-s + (0.707 − 0.707i)12-s + i·13-s + (−0.707 − 0.707i)14-s + 16-s i·18-s + i·19-s − 21-s + (0.707 − 0.707i)22-s + ⋯
L(s,χ)  = 1  + 2-s + (0.707 − 0.707i)3-s + 4-s + (0.707 − 0.707i)6-s + (−0.707 − 0.707i)7-s + 8-s i·9-s + (0.707 − 0.707i)11-s + (0.707 − 0.707i)12-s + i·13-s + (−0.707 − 0.707i)14-s + 16-s i·18-s + i·19-s − 21-s + (0.707 − 0.707i)22-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.564 - 0.825i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.564 - 0.825i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(85\)    =    \(5 \cdot 17\)
\( \varepsilon \)  =  $0.564 - 0.825i$
motivic weight  =  \(0\)
character  :  $\chi_{85} (77, \cdot )$
Sato-Tate  :  $\mu(8)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 85,\ (1:\ ),\ 0.564 - 0.825i)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.306900455 - 1.744140732i$
$L(\frac12,\chi)$  $\approx$  $3.306900455 - 1.744140732i$
$L(\chi,1)$  $\approx$  2.234973436 - 0.7061374073i
$L(1,\chi)$  $\approx$  2.234973436 - 0.7061374073i

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

−30.803420747922941451576663099550, −30.00601408314237265171870599325, −28.527060122286135654851993800105, −27.64446895151106455123147691495, −25.98781509983757350214017751355, −25.363351680766149564240474387558, −24.43181994537158636163895765089, −22.74065357076088524610104776546, −22.19377360722073305303821461771, −21.13085135317110113969094418398, −20.01922902400411570913054640596, −19.32801183786490852736557904533, −17.316175513184322518768937038895, −15.713283978378061598142647827279, −15.40134370978321734058756182921, −14.13992717643873104463026589239, −13.03871980294941738361869021479, −11.885869098630981034143982957880, −10.38847760440593726111994776272, −9.24840086776312417597462176282, −7.65719490785394029755611514586, −6.09872388087155823967100055449, −4.748148868584009453015508750209, −3.457956661405640418945804152221, −2.32968474890305295936909797420, 1.43644379061914263667458918564, 3.10861623608053527279576182566, 4.13876931827194811027384660645, 6.249327812753090324676014694353, 6.9590735885187606061305381916, 8.4471260596983335143522506459, 10.10143157917135555912863423860, 11.72991286987807773182716123822, 12.68909151167478362807786928784, 13.94675993836388078814453022064, 14.27916791656150432749130452834, 15.9420948523297909841040665654, 16.94475797560516927502120951337, 18.835342328822274740964565326255, 19.662288220205569546105109941727, 20.59223568789344073716795839729, 21.79315992125735158432127600198, 23.02876699311268419994810879075, 23.91964861408058195547481836586, 24.83026959507889161801752512002, 25.826885276448170752436845670471, 26.80198354864851235451183767947, 28.82899249986115262470030821011, 29.56352946201280617642286049480, 30.37949726527011483859753243971

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