Properties

Degree $1$
Conductor $97$
Sign $-0.691 - 0.722i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.691 - 0.722i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.691 - 0.722i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.691 - 0.722i$
Motivic weight: \(0\)
Character: $\chi_{97} (55, \cdot )$
Sato-Tate group: $\mu(32)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ -0.691 - 0.722i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.6086008077 - 1.425740799i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.6086008077 - 1.425740799i\)
\(L(\chi,1)\) \(\approx\) \(0.8870100423 - 0.6686603712i\)
\(L(1,\chi)\) \(\approx\) \(0.8870100423 - 0.6686603712i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.2189637164987893716821950809, −29.48740337734195616739928922818, −28.00138922777496658441357042626, −27.0186297701899009960513824478, −25.91105615870423809188566948368, −24.7965445000924286938847030664, −23.94957717000854766361256828356, −23.09762913901626800669662862703, −21.88050206098978662260422883317, −21.47129487657514236608330329575, −19.35973288001056513616053459569, −17.843811188590661508663446714086, −17.45348974702473481001637269732, −16.61980832571858420455382233872, −14.96532179877829142294921674448, −14.09470081099802042535156712638, −13.040087842924821092273241533018, −11.74632936226416115393509058706, −10.501994714433026295872757048312, −8.95653211635455138183288884644, −7.1754634504260827658704209177, −6.65515665782547015819276542057, −5.32352820485880952303234366069, −4.197998222469448900387601206607, −1.72204063813781357952877903506, 0.72185602346934564771666374972, 2.222077807689065151360441341340, 4.2863203871060639910658048605, 5.25096154335157294816944046047, 6.18894146420669907248910024317, 8.70397275365782962477010841803, 9.753105412582440752448195790426, 10.86412032558693357171708039630, 11.9134556166032867433383709356, 12.75165521890385923932957178573, 14.14537489181945716317991510330, 15.25197785403910595761117251694, 16.9480878770674192208890694827, 17.69753532576887060221827076088, 18.73319322116276110658059670060, 20.36725990943556308468918939087, 21.1274337520767795804976302059, 22.03217211365125005093935328295, 22.71965464298561476891891842846, 24.21990518545733217942320411298, 24.833494492935833671467959362036, 26.96511280334682524085770645959, 27.67892945708414281906633694160, 28.46529813479297390658569843034, 29.48406958179631613498992331773

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