↑

Properties

Label 1-97-97.85-r0-0-0
Degree $1$
Conductor $97$
Sign $-0.393 - 0.919i$
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  + (0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s − i·4-s + (0.923 − 0.382i)5-s − 6-s + (0.923 + 0.382i)7-s + (−0.707 − 0.707i)8-s + i·9-s + (0.382 − 0.923i)10-s + (−0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s + (−0.923 + 0.382i)13-s + (0.923 − 0.382i)14-s + (−0.923 − 0.382i)15-s − 16-s + (−0.923 + 0.382i)17-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s − i·4-s + (0.923 − 0.382i)5-s − 6-s + (0.923 + 0.382i)7-s + (−0.707 − 0.707i)8-s + i·9-s + (0.382 − 0.923i)10-s + (−0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s + (−0.923 + 0.382i)13-s + (0.923 − 0.382i)14-s + (−0.923 − 0.382i)15-s − 16-s + (−0.923 + 0.382i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.393 - 0.919i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ -0.393 - 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7195034467 - 1.090196433i\)
\(L(\frac12)\) \(\approx\) \(0.7195034467 - 1.090196433i\)
\(L(1)\) \(\approx\) \(1.018368144 - 0.8371583131i\)
\(L(1)\) \(\approx\) \(1.018368144 - 0.8371583131i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 + (0.923 + 0.382i)T \)
11 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (-0.923 + 0.382i)T \)
17 \( 1 + (-0.923 + 0.382i)T \)
19 \( 1 + (0.923 - 0.382i)T \)
23 \( 1 + (0.382 + 0.923i)T \)
29 \( 1 + (0.382 + 0.923i)T \)
31 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (-0.382 - 0.923i)T \)
43 \( 1 - iT \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (0.382 - 0.923i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.923 + 0.382i)T \)
71 \( 1 + (-0.382 + 0.923i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + (0.923 - 0.382i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
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

−30.4888900310269153691159329209, −29.476377316893449692228653046388, −28.46114664147683213436409187617, −26.80353105001990405966987786478, −26.53571439731544595717508101066, −25.00862908132174696022558976000, −24.16658726965734530026954579927, −22.83104368270275699712865322250, −22.32015302673595627292643128891, −21.101808043423506605071212955009, −20.5859880596814999919045202855, −17.879509874591297242872622886690, −17.66320752700602947486089483433, −16.48468284714363968708170510861, −15.21332886593742392135639936240, −14.47003046637500504491345148376, −13.2288463446882322166052856871, −11.90777282512552211333410301380, −10.68287637843254956573041086150, −9.51780530270899800744507429658, −7.73119254724387419455558449843, −6.49262156995221291145048925454, −5.20914030715717297820385352924, −4.5378882672659743431410016728, −2.614512229051286648511670053163, 1.41679223254449677072988432355, 2.53071296480721811866723847296, 4.97545871934902319279881368531, 5.432966084220273174244268696726, 6.86879751724370292912608365322, 8.71562336282065583657317511106, 10.27399631768084377518061281693, 11.33962213424888842159407576014, 12.280228323891796535920251178687, 13.40579377537937880255371577283, 14.09427568294314108985710501487, 15.68395885387361440964850244528, 17.28300416571397953599215982583, 18.09128135776569827477593326422, 19.147085488686319935081795904244, 20.456318616819286740330109896104, 21.71814322319928272393461316171, 22.01106167052246786614851879134, 23.72743768575642131837322576264, 24.21579549095195024553292587952, 25.0631042817907373943988270597, 26.98016684680569563954355110944, 28.25127910528354103067603415378, 28.980861237292382987403711406802, 29.564796266244381009029956536627

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