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Properties

Label 1-97-97.68-r1-0-0
Degree $1$
Conductor $97$
Sign $-0.0806 + 0.996i$
Analytic cond. $10.4240$
Root an. cond. $10.4240$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.0806 + 0.996i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ -0.0806 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04338401773 - 0.04703711715i\)
\(L(\frac12)\) \(\approx\) \(-0.04338401773 - 0.04703711715i\)
\(L(1)\) \(\approx\) \(0.3341164959 - 0.1993971205i\)
\(L(1)\) \(\approx\) \(0.3341164959 - 0.1993971205i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (-0.793 - 0.608i)T \)
3 \( 1 + (-0.991 + 0.130i)T \)
5 \( 1 + (-0.659 - 0.751i)T \)
7 \( 1 + (-0.321 - 0.946i)T \)
11 \( 1 + (-0.608 - 0.793i)T \)
13 \( 1 + (0.751 - 0.659i)T \)
17 \( 1 + (-0.946 - 0.321i)T \)
19 \( 1 + (-0.980 + 0.195i)T \)
23 \( 1 + (0.896 - 0.442i)T \)
29 \( 1 + (-0.0654 + 0.997i)T \)
31 \( 1 + (0.130 + 0.991i)T \)
37 \( 1 + (0.442 - 0.896i)T \)
41 \( 1 + (0.997 + 0.0654i)T \)
43 \( 1 + (-0.965 - 0.258i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (-0.608 + 0.793i)T \)
59 \( 1 + (-0.896 - 0.442i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.195 + 0.980i)T \)
71 \( 1 + (0.997 - 0.0654i)T \)
73 \( 1 + (0.258 - 0.965i)T \)
79 \( 1 + (0.923 + 0.382i)T \)
83 \( 1 + (0.321 - 0.946i)T \)
89 \( 1 + (-0.382 + 0.923i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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