Properties

Degree 1
Conductor 97
Sign $-0.0806 + 0.996i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.793 − 0.608i)2-s + (−0.991 + 0.130i)3-s + (0.258 + 0.965i)4-s + (−0.659 − 0.751i)5-s + (0.866 + 0.5i)6-s + (−0.321 − 0.946i)7-s + (0.382 − 0.923i)8-s + (0.965 − 0.258i)9-s + (0.0654 + 0.997i)10-s + (−0.608 − 0.793i)11-s + (−0.382 − 0.923i)12-s + (0.751 − 0.659i)13-s + (−0.321 + 0.946i)14-s + (0.751 + 0.659i)15-s + (−0.866 + 0.5i)16-s + (−0.946 − 0.321i)17-s + ⋯
L(s,χ)  = 1  + (−0.793 − 0.608i)2-s + (−0.991 + 0.130i)3-s + (0.258 + 0.965i)4-s + (−0.659 − 0.751i)5-s + (0.866 + 0.5i)6-s + (−0.321 − 0.946i)7-s + (0.382 − 0.923i)8-s + (0.965 − 0.258i)9-s + (0.0654 + 0.997i)10-s + (−0.608 − 0.793i)11-s + (−0.382 − 0.923i)12-s + (0.751 − 0.659i)13-s + (−0.321 + 0.946i)14-s + (0.751 + 0.659i)15-s + (−0.866 + 0.5i)16-s + (−0.946 − 0.321i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.0806 + 0.996i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.0806 + 0.996i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(97\)
\( \varepsilon \)  =  $-0.0806 + 0.996i$
motivic weight  =  \(0\)
character  :  $\chi_{97} (68, \cdot )$
Sato-Tate  :  $\mu(96)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 97,\ (1:\ ),\ -0.0806 + 0.996i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.04338401773 - 0.04703711715i$
$L(\frac12,\chi)$  $\approx$  $-0.04338401773 - 0.04703711715i$
$L(\chi,1)$  $\approx$  0.3341164959 - 0.1993971205i
$L(1,\chi)$  $\approx$  0.3341164959 - 0.1993971205i

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

−30.520117451427756658682642803696, −29.132068348248399581610483792969, −28.32434885755007406139209201617, −27.661096756793604619252495336651, −26.43879981994147820033009622263, −25.58504528055009094443542955311, −24.28410186990780804955049837699, −23.32588375857938370526294981147, −22.64138166400182473333640310756, −21.26749422612753923398654466630, −19.464846663321430237415720207622, −18.645605214319286336806363008457, −17.96350940034899676656435736221, −16.76450408325451797425623687780, −15.47498729894627278430413727905, −15.20451479855838614064666875656, −13.1440922631505121416552497857, −11.62182108229348553044960246133, −10.88317128625799493752201899966, −9.62080509389749482757063527162, −8.15279754167371024084940334982, −6.83255726647034105922079482669, −6.12377456069691698299251998644, −4.58299354792804649594842743982, −2.13934491288993443037973495126, 0.05214442837044965197291254473, 1.0330799932755172920037071993, 3.487040546238733594763530952905, 4.72037332921633168095661100818, 6.57042420194290712125195537856, 7.87836099864399646073289039789, 9.081401529056403721104918283072, 10.75571430210962991852118550573, 10.97719804846978891645408624542, 12.57511457385283560559051105002, 13.17972645753975583201819700020, 15.75440399274722615561613960385, 16.37198264937823202496606215127, 17.26999503256541444038675689372, 18.372114617449484911292578092089, 19.53116629532647191569845329946, 20.54545964641241356432628561361, 21.45571293058817729059547374056, 22.85811280233235512411613853391, 23.64813694742603124292981091024, 24.9180850911216886026070354731, 26.497670801765726373949066759992, 27.174159981005439794893913785305, 28.07473923980227899995593420900, 28.99171734333430458808618306691

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