Properties

Degree 1
Conductor 97
Sign $0.112 + 0.993i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.923 + 0.382i)2-s + (0.382 + 0.923i)3-s + (0.707 + 0.707i)4-s + (0.980 + 0.195i)5-s + i·6-s + (0.980 − 0.195i)7-s + (0.382 + 0.923i)8-s + (−0.707 + 0.707i)9-s + (0.831 + 0.555i)10-s + (−0.382 − 0.923i)11-s + (−0.382 + 0.923i)12-s + (0.195 − 0.980i)13-s + (0.980 + 0.195i)14-s + (0.195 + 0.980i)15-s + i·16-s + (0.195 − 0.980i)17-s + ⋯
L(s,χ)  = 1  + (0.923 + 0.382i)2-s + (0.382 + 0.923i)3-s + (0.707 + 0.707i)4-s + (0.980 + 0.195i)5-s + i·6-s + (0.980 − 0.195i)7-s + (0.382 + 0.923i)8-s + (−0.707 + 0.707i)9-s + (0.831 + 0.555i)10-s + (−0.382 − 0.923i)11-s + (−0.382 + 0.923i)12-s + (0.195 − 0.980i)13-s + (0.980 + 0.195i)14-s + (0.195 + 0.980i)15-s + i·16-s + (0.195 − 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.112 + 0.993i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.112 + 0.993i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(97\)
\( \varepsilon \)  =  $0.112 + 0.993i$
motivic weight  =  \(0\)
character  :  $\chi_{97} (28, \cdot )$
Sato-Tate  :  $\mu(32)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 97,\ (1:\ ),\ 0.112 + 0.993i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(2.980835797 + 2.661750863i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(2.980835797 + 2.661750863i\)
\(L(\chi,1)\)  \(\approx\)  \(2.106063090 + 1.193751533i\)
\(L(1,\chi)\)  \(\approx\)  \(2.106063090 + 1.193751533i\)

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

−29.91268743994250901329811206128, −28.761417419429080496258097062757, −28.10560767780696410921400236729, −25.964565074858696217375703273944, −25.259880608169779543952618609112, −24.089826968182005202662322486264, −23.67510685170426465632675081577, −22.13546427381257977352021473602, −21.01394227165092313766946576802, −20.45747824376785486386842741738, −19.041777871732185850312000187025, −18.065942260300867697018241868580, −16.88891154209956414512460616350, −14.90997159359676214306334502883, −14.34627535211527155049197063022, −13.19794453493343971934356759371, −12.43825703672123761913289390480, −11.20859610866081099909515406635, −9.771067050378553558055565209238, −8.232727056878134552391175683440, −6.72298092759049698915138994258, −5.68560076878287495975150265142, −4.2558037470489967281145463979, −2.17924111868267550516856221321, −1.70529206117758640916970959428, 2.285785229023347035260081532376, 3.56037114603839835822668426306, 5.07440676559678244680498158760, 5.77589563200868465432660290663, 7.64199752504461282623580195772, 8.81913017478918558840266366244, 10.491679215875468917244859305689, 11.25029995669998960498459386118, 13.11470849622339620981464100119, 14.05440909876664303800880468441, 14.752655357846972803644228485364, 15.93772332154581156788348839561, 16.96758633431920990412417466148, 18.07640137550021440817497167905, 20.08568137383058411081620618946, 20.99028765320205735009089299125, 21.58733717266181805599573515680, 22.50860796498942118582350631460, 23.82725266610040045942438695297, 24.97596426370069818965361663467, 25.728838325946029941658539433477, 26.76546496153567452798606789492, 27.848098736154156122873537047893, 29.44272876589120637000904057379, 30.13546605359060230826593640365

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