Properties

Degree 1
Conductor 97
Sign $0.860 + 0.508i$
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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.860 + 0.508i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.860 + 0.508i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(97\)
\( \varepsilon \)  =  $0.860 + 0.508i$
motivic weight  =  \(0\)
character  :  $\chi_{97} (48, \cdot )$
Sato-Tate  :  $\mu(48)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 97,\ (0:\ ),\ 0.860 + 0.508i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.718914927 + 0.4698791485i$
$L(\frac12,\chi)$  $\approx$  $1.718914927 + 0.4698791485i$
$L(\chi,1)$  $\approx$  1.721483205 + 0.2979663345i
$L(1,\chi)$  $\approx$  1.721483205 + 0.2979663345i

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

−30.03405465929483439965684813943, −28.91142815378259587529120256915, −28.81345396440999593725429982349, −26.346293245933125943140762560505, −25.429516366410467005989594117585, −24.808004143388127966644176538813, −23.72408603917189047058564501841, −22.98313068105265669746481783605, −21.60594664584425009108297557729, −20.622385832847689033381207060724, −19.61978951862054658071571072969, −18.35171651479661812373943623546, −16.88849380085109803367527478644, −16.14041791120127677499501349651, −14.51578649878212733947262102252, −13.64211627611770134496016505122, −12.52984818136160952043458517719, −12.26176186562112670965789359240, −10.1388986007963550367917147334, −8.57370387947060627174618124737, −7.26721893246254251997102134289, −6.15363932376450579208177009024, −5.07849153287870071247515871787, −3.18536992412232966565117024243, −1.89999181132362608181966129234, 2.7635634161126596312508867674, 3.267574842240073204288825746048, 5.05187189119031285576852841266, 6.009718988875711604476225360068, 7.53140386272137829602768694341, 9.79290349419283269696941203459, 10.2168490883041642058357458295, 11.45312095747561503400474483700, 13.10880867340335101108887055888, 13.88946311788484686549824356583, 15.12915023486368391376323917241, 15.772741736791380048153914595122, 17.084530575123921464286724575846, 18.84946479188500875915288850987, 19.9033087163886234975741334330, 20.98034482744242178529139534905, 21.909564934651511601103340024439, 22.52047552073711727914360427481, 23.53374614986010358307589351761, 25.27058874086582269780213689468, 25.81533073108637366715382574542, 26.9469937660029743460492464900, 28.41715571963721873819554387325, 29.29051592224269356974916397763, 30.17861796004479086843573706832

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