↑

Properties

Label 1-97-97.16-r0-0-0
Degree $1$
Conductor $97$
Sign $-0.732 + 0.680i$
Analytic cond. $0.450466$
Root an. cond. $0.450466$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.5 + 0.866i)6-s + (0.866 − 0.5i)7-s − 8-s + (−0.5 + 0.866i)9-s + (−0.866 − 0.5i)10-s + (0.5 − 0.866i)11-s − 12-s + (−0.866 + 0.5i)13-s + (0.866 + 0.5i)14-s + (−0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.5 + 0.866i)6-s + (0.866 − 0.5i)7-s − 8-s + (−0.5 + 0.866i)9-s + (−0.866 − 0.5i)10-s + (0.5 − 0.866i)11-s − 12-s + (−0.866 + 0.5i)13-s + (0.866 + 0.5i)14-s + (−0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.732 + 0.680i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ -0.732 + 0.680i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4677505657 + 1.189880019i\)
\(L(\frac12)\) \(\approx\) \(0.4677505657 + 1.189880019i\)
\(L(1)\) \(\approx\) \(0.8815841335 + 0.9555284392i\)
\(L(1)\) \(\approx\) \(0.8815841335 + 0.9555284392i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 - iT \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 - T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.01397245699476148505836545369, −28.81174107663063101523962623386, −27.73915872843710074958947934066, −27.023478255443824853226226655143, −25.05156333545462987138164071765, −24.44348815941167596724367144931, −23.3666323830797903551365381990, −22.5274679334358383217672386357, −20.7946723232600903166911475452, −20.35863428393997370435234990210, −19.202947761907831638169474432565, −18.45532925612415099888266389012, −17.1499607132237091684777830389, −14.97385020675207177075954376964, −14.66678741519342130534859612404, −13.07873726029619155306720896703, −12.116235203070772746398432117619, −11.64858728111833390735168861661, −9.75991508803278812903991443906, −8.48669769541821797283831931992, −7.39050488268241658681530192067, −5.48438333238163522695330321622, −4.19423728347773412412904334112, −2.66293989880087845544542945206, −1.2816232539697457611774498157, 3.12417867434519404051209446344, 4.137223114266321929618509679911, 5.19234917150558938507332336985, 7.05329314610907417805561632998, 8.02731472595670775968205175911, 9.095492477146754628558526480200, 10.82412894002183141930681099644, 11.85261574750081415520675102147, 13.68877057560360251853010119733, 14.588599604475501344414181575201, 15.17283167422813241441285854846, 16.43974254151170399346862023749, 17.20219307278718473282570090890, 18.90348172661814489566861281651, 20.03423499151736876453700104405, 21.43036086656760905167610971599, 21.99979771917604331817624358799, 23.329380970395639583254243300676, 24.123527023722641066650583318753, 25.32103289062993772563147593977, 26.679905566545335466616919694070, 26.852904079403234133218575974813, 27.85937094581148857035704900857, 30.001041217170997156385922062057, 30.73906478286703626437801564077

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