Properties

Degree 1
Conductor 97
Sign $-0.535 - 0.844i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.130 + 0.991i)2-s + (0.608 + 0.793i)3-s + (−0.965 − 0.258i)4-s + (−0.321 + 0.946i)5-s + (−0.866 + 0.5i)6-s + (−0.659 + 0.751i)7-s + (0.382 − 0.923i)8-s + (−0.258 + 0.965i)9-s + (−0.896 − 0.442i)10-s + (0.991 − 0.130i)11-s + (−0.382 − 0.923i)12-s + (−0.946 − 0.321i)13-s + (−0.659 − 0.751i)14-s + (−0.946 + 0.321i)15-s + (0.866 + 0.5i)16-s + (0.751 − 0.659i)17-s + ⋯
L(s,χ)  = 1  + (−0.130 + 0.991i)2-s + (0.608 + 0.793i)3-s + (−0.965 − 0.258i)4-s + (−0.321 + 0.946i)5-s + (−0.866 + 0.5i)6-s + (−0.659 + 0.751i)7-s + (0.382 − 0.923i)8-s + (−0.258 + 0.965i)9-s + (−0.896 − 0.442i)10-s + (0.991 − 0.130i)11-s + (−0.382 − 0.923i)12-s + (−0.946 − 0.321i)13-s + (−0.659 − 0.751i)14-s + (−0.946 + 0.321i)15-s + (0.866 + 0.5i)16-s + (0.751 − 0.659i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.535 - 0.844i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.535 - 0.844i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(97\)
\( \varepsilon \)  =  $-0.535 - 0.844i$
motivic weight  =  \(0\)
character  :  $\chi_{97} (74, \cdot )$
Sato-Tate  :  $\mu(96)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 97,\ (1:\ ),\ -0.535 - 0.844i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.5007805212 + 0.9103191966i$
$L(\frac12,\chi)$  $\approx$  $-0.5007805212 + 0.9103191966i$
$L(\chi,1)$  $\approx$  0.4327643782 + 0.7891595847i
$L(1,\chi)$  $\approx$  0.4327643782 + 0.7891595847i

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

−29.337550469962682389316073404982, −28.333970834475488489815561007426, −27.19147812071252041668322798782, −26.18139108335894466103626832394, −24.99624229576741929670478297878, −23.79523766067362379550273954216, −22.96128280634151946303885828612, −21.49018893631461038224751678893, −20.3328067269715166758720555062, −19.58258012949740616573866711276, −19.11939999559753732778542874981, −17.41600564499365622613966872587, −16.73576139077717674548406202272, −14.65789507438100169026456664424, −13.63064089136509982303859908875, −12.50252768417405378777752060397, −12.078535492222867367119993609098, −10.228297758041830585256935549, −9.08477248377812194824059270599, −8.16236779793808604188456029630, −6.70447514552091122223900939071, −4.53546871803094386595414827693, −3.40496339898200625006313411477, −1.72509960138381734282043492139, −0.45900167332815919977253205206, 2.86422939209670167698324337761, 4.11105996142828458359615897086, 5.66642128308457712046842771688, 6.95859341317668253054966963881, 8.20281459958743733499238314837, 9.43159802170147471758568829431, 10.19873080575857526141633311200, 11.98349403490263694162302979332, 13.7272088676317509545729015810, 14.71804801978283526456492015159, 15.324221224667356279686251404029, 16.33894094148402779812518764103, 17.5373378084162218457722269227, 19.15080875788367522710877017520, 19.38528411142801853122213650730, 21.44477448996797089957059023044, 22.33177429042708068567018566569, 22.95429413547714761538252199371, 24.68926321917740835153219485722, 25.46629868345025844091705685534, 26.216426421790392779482087534030, 27.37087104948002432184232098112, 27.71923165111849132903294939699, 29.575590133281101985097274137021, 30.93027059972099320272810108155

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