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Properties

Label 1-2153-2153.33-r0-0-0
Degree $1$
Conductor $2153$
Sign $0.998 - 0.0513i$
Analytic cond. $9.99848$
Root an. cond. $9.99848$
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  + (−0.666 + 0.745i)2-s + (−0.850 + 0.526i)3-s + (−0.110 − 0.993i)4-s + (0.965 − 0.259i)5-s + (0.174 − 0.984i)6-s + (−0.546 − 0.837i)7-s + (0.814 + 0.580i)8-s + (0.445 − 0.895i)9-s + (−0.450 + 0.892i)10-s + (−0.653 + 0.756i)11-s + (0.617 + 0.786i)12-s + (−0.662 + 0.749i)13-s + (0.988 + 0.151i)14-s + (−0.684 + 0.729i)15-s + (−0.975 + 0.220i)16-s + (−0.995 − 0.0991i)17-s + ⋯
L(s)  = 1  + (−0.666 + 0.745i)2-s + (−0.850 + 0.526i)3-s + (−0.110 − 0.993i)4-s + (0.965 − 0.259i)5-s + (0.174 − 0.984i)6-s + (−0.546 − 0.837i)7-s + (0.814 + 0.580i)8-s + (0.445 − 0.895i)9-s + (−0.450 + 0.892i)10-s + (−0.653 + 0.756i)11-s + (0.617 + 0.786i)12-s + (−0.662 + 0.749i)13-s + (0.988 + 0.151i)14-s + (−0.684 + 0.729i)15-s + (−0.975 + 0.220i)16-s + (−0.995 − 0.0991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2153\)
Sign: $0.998 - 0.0513i$
Analytic conductor: \(9.99848\)
Root analytic conductor: \(9.99848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2153} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2153,\ (0:\ ),\ 0.998 - 0.0513i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6215822858 + 0.01597228336i\)
\(L(\frac12)\) \(\approx\) \(0.6215822858 + 0.01597228336i\)
\(L(1)\) \(\approx\) \(0.5568102689 + 0.1603785709i\)
\(L(1)\) \(\approx\) \(0.5568102689 + 0.1603785709i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2153 \( 1 \)
good2 \( 1 + (-0.666 + 0.745i)T \)
3 \( 1 + (-0.850 + 0.526i)T \)
5 \( 1 + (0.965 - 0.259i)T \)
7 \( 1 + (-0.546 - 0.837i)T \)
11 \( 1 + (-0.653 + 0.756i)T \)
13 \( 1 + (-0.662 + 0.749i)T \)
17 \( 1 + (-0.995 - 0.0991i)T \)
19 \( 1 + (0.271 - 0.962i)T \)
23 \( 1 + (0.879 + 0.476i)T \)
29 \( 1 + (-0.814 + 0.580i)T \)
31 \( 1 + (-0.977 - 0.208i)T \)
37 \( 1 + (0.771 - 0.635i)T \)
41 \( 1 + (0.984 - 0.174i)T \)
43 \( 1 + (0.688 + 0.725i)T \)
47 \( 1 + (-0.721 + 0.692i)T \)
53 \( 1 + (-0.197 - 0.980i)T \)
59 \( 1 + (0.565 - 0.824i)T \)
61 \( 1 + (-0.895 - 0.445i)T \)
67 \( 1 + (0.998 - 0.0466i)T \)
71 \( 1 + (-0.551 + 0.834i)T \)
73 \( 1 + (0.565 + 0.824i)T \)
79 \( 1 + (-0.960 + 0.276i)T \)
83 \( 1 + (0.145 + 0.989i)T \)
89 \( 1 + (0.999 + 0.0350i)T \)
97 \( 1 + (0.919 + 0.392i)T \)
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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

−19.56784849506594348058740142806, −18.775995334819579715070575631336, −18.415535671584369749264788884834, −17.8598870394931664845805859370, −17.052115699194100817772274937308, −16.53970699318264202706958004076, −15.73129730553110314308827120293, −14.69718836208559108965402394524, −13.41533708609531665845607075305, −13.09252831585523694784886453169, −12.48544106606208623500465934738, −11.667659691425480746307082981793, −10.80200899004687619062291983514, −10.41773349487000717951261561413, −9.53405657137929731400423459268, −8.846601996117896150533940387975, −7.87539085851425094221017843282, −7.122902209672670310032161119794, −6.13220880421059881681145335217, −5.62992298489166079109267523722, −4.71617115856544965035339917271, −3.27894521422832248301266184541, −2.50428553892201611627008356034, −1.93583865228041373011601386083, −0.73769072474361802303032353915, 0.42121341251268273803505113026, 1.530626632594171558502640568765, 2.55495850052504804652499366399, 4.14447954923781926155359810875, 4.870552315960851790228875730, 5.389885787920511127551100399844, 6.39709191312249028925126229643, 6.95496183301179149814175779934, 7.51425805905170807447768338230, 9.058284102394443402831089342107, 9.491415103665006979631194148419, 9.87083448390958299082187746638, 11.00929023311204525538451204088, 11.103174846765503028368051796899, 12.87861627711266362270621186181, 13.00816911935612707911876056961, 14.18974669945466597921925085597, 14.829036022873149302055482716730, 15.77604429715220987540652179754, 16.27575359617168058746558565654, 16.95704989518945556884584812013, 17.57811326536390972624607555183, 17.87123951417616160134772428446, 18.814832275221949714459238562081, 19.83766692548217204788146606253

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