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Properties

Label 1-97-97.45-r1-0-0
Degree $1$
Conductor $97$
Sign $-0.964 - 0.265i$
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  + (0.923 − 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)  = 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.964 - 0.265i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ -0.964 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2675172130 - 1.981919722i\)
\(L(\frac12)\) \(\approx\) \(0.2675172130 - 1.981919722i\)
\(L(1)\) \(\approx\) \(1.088694245 - 0.9961987866i\)
\(L(1)\) \(\approx\) \(1.088694245 - 0.9961987866i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (0.923 - 0.382i)T \)
3 \( 1 + (0.382 - 0.923i)T \)
5 \( 1 + (-0.980 + 0.195i)T \)
7 \( 1 + (-0.980 - 0.195i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 + (-0.195 - 0.980i)T \)
17 \( 1 + (-0.195 - 0.980i)T \)
19 \( 1 + (0.980 - 0.195i)T \)
23 \( 1 + (0.831 + 0.555i)T \)
29 \( 1 + (0.831 + 0.555i)T \)
31 \( 1 + (-0.923 - 0.382i)T \)
37 \( 1 + (-0.555 - 0.831i)T \)
41 \( 1 + (0.555 - 0.831i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (-0.382 - 0.923i)T \)
59 \( 1 + (-0.831 + 0.555i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.195 - 0.980i)T \)
71 \( 1 + (0.555 + 0.831i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (0.923 + 0.382i)T \)
83 \( 1 + (0.980 - 0.195i)T \)
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

−31.06085213609323666824341577220, −29.25421210826085104294380829003, −28.34980047947732709748713330966, −26.67867385098324347158678735112, −26.38815526864615942118741609322, −25.03188555291813449348104527866, −23.924551194956057780377827501, −22.89926863726792438254080762773, −21.968813974980613568724076481409, −21.103358920483352860731660273764, −19.90029368051646938512050446930, −19.08926328821187404404134907953, −16.70972303684768942174421056343, −16.16153295990171352879712386347, −15.35334405418141500149056384770, −14.26168718023019510237971204961, −13.067125735452945069390265021518, −11.81497378374550240817261049848, −10.73337410627941408886467197527, −9.033556700182783992889660328360, −7.94289087873776143206990997327, −6.41445678069028843313126825874, −4.97433212923751745734861047088, −3.79399034529981048799996897495, −2.9434207084666478800855636092, 0.61477968900466800291265328445, 2.65408091406076051856471948424, 3.533716519071703002833503235626, 5.27119633463835604146591735396, 6.94626422355970689205320077572, 7.48928708672193087283743523997, 9.482104566502726113457000056476, 10.97556328428863135870439841150, 12.26282795339107465985026281046, 12.8213850148097216199696691864, 13.97771616005178940221458747824, 15.19950257282331412950818074216, 16.00506156018280625501758197811, 17.950554030859724048766316858335, 19.13453530103425917870386594384, 19.9255602834261178260951951496, 20.54822655452501898983390694601, 22.585961812944585301139243221022, 22.890617993631211778407041836798, 23.92879978817211262406326730861, 25.0166393689890844605152443159, 25.91976572195680118110714879736, 27.43405732566148663485726610801, 28.82642957353779814449253659219, 29.50300968439320216596479290271

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