Properties

Degree 1
Conductor 97
Sign $0.998 + 0.0517i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.965 − 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 − 0.5i)4-s + (−0.608 − 0.793i)5-s + (0.5 + 0.866i)6-s + (0.991 + 0.130i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (−0.793 − 0.608i)10-s + (−0.965 − 0.258i)11-s + (0.707 + 0.707i)12-s + (0.608 + 0.793i)13-s + (0.991 − 0.130i)14-s + (0.608 − 0.793i)15-s + (0.5 − 0.866i)16-s + (−0.991 + 0.130i)17-s + ⋯
L(s,χ)  = 1  + (0.965 − 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 − 0.5i)4-s + (−0.608 − 0.793i)5-s + (0.5 + 0.866i)6-s + (0.991 + 0.130i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (−0.793 − 0.608i)10-s + (−0.965 − 0.258i)11-s + (0.707 + 0.707i)12-s + (0.608 + 0.793i)13-s + (0.991 − 0.130i)14-s + (0.608 − 0.793i)15-s + (0.5 − 0.866i)16-s + (−0.991 + 0.130i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.998 + 0.0517i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.998 + 0.0517i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(97\)
\( \varepsilon \)  =  $0.998 + 0.0517i$
motivic weight  =  \(0\)
character  :  $\chi_{97} (49, \cdot )$
Sato-Tate  :  $\mu(48)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 97,\ (0:\ ),\ 0.998 + 0.0517i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.747529459 + 0.04528578014i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.747529459 + 0.04528578014i\)
\(L(\chi,1)\)  \(\approx\)  \(1.715567167 + 0.02660170940i\)
\(L(1,\chi)\)  \(\approx\)  \(1.715567167 + 0.02660170940i\)

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

−30.3721687580719247025844057860, −29.60739375722326355444804952544, −28.21848167221116012654042440181, −26.59841285296253857434425469865, −25.73910054264005996932855718249, −24.63558483658326344606940319009, −23.61383243853434414837780547310, −23.20136955505719293596394397208, −21.861996754832148512490057972999, −20.55120171777560888300209926396, −19.73114496794814755700373138376, −18.26059396461625821640917734105, −17.50634764060209825252828298811, −15.57948263802039994958803038056, −14.953095727393452662089121971353, −13.739364770798918585270609321546, −12.945688418124425393808933025552, −11.53269195041163359957693218431, −10.90654355827872297482482761973, −8.228086845390080192773476600405, −7.54527889367247078797641978556, −6.45064817478947638830467534786, −4.97493867965834776328255690590, −3.33255020608720458199724382089, −2.12416512879201328176508009709, 2.11608194739238174434185996990, 3.89762355216051094627785199579, 4.63889110241892541940235687206, 5.7414785725942689593431267215, 7.847813569541507806710771697255, 8.96031140042146818164921751421, 10.71418942157101797343277020207, 11.36157169621747228063100482471, 12.70975286403331488191126234851, 13.97498149660048497670573796191, 15.00798644508859618055574919333, 15.911954442935816000774531925343, 16.74641283621266643446728902844, 18.71694395542793841583243019343, 20.11343963756053627135711820985, 20.829670171902801572337805497996, 21.4071137672641415987828002259, 22.718770706455180997684646466472, 23.82821789446622455767901094093, 24.52192500462472943301609965273, 25.94995067979923822628162380188, 27.16716201149786016375868918384, 28.20071492091146746929167942120, 28.855616850341338047637022983473, 30.56147903294024453094208453166

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