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Properties

Label 1-101-101.18-r1-0-0
Degree $1$
Conductor $101$
Sign $-0.329 - 0.943i$
Analytic cond. $10.8539$
Root an. cond. $10.8539$
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  + (−0.770 + 0.637i)2-s + (0.844 − 0.535i)3-s + (0.187 − 0.982i)4-s + (−0.637 + 0.770i)5-s + (−0.309 + 0.951i)6-s + (−0.998 − 0.0627i)7-s + (0.481 + 0.876i)8-s + (0.425 − 0.904i)9-s − i·10-s + (0.904 + 0.425i)11-s + (−0.368 − 0.929i)12-s + (−0.0627 − 0.998i)13-s + (0.809 − 0.587i)14-s + (−0.125 + 0.992i)15-s + (−0.929 − 0.368i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (−0.770 + 0.637i)2-s + (0.844 − 0.535i)3-s + (0.187 − 0.982i)4-s + (−0.637 + 0.770i)5-s + (−0.309 + 0.951i)6-s + (−0.998 − 0.0627i)7-s + (0.481 + 0.876i)8-s + (0.425 − 0.904i)9-s − i·10-s + (0.904 + 0.425i)11-s + (−0.368 − 0.929i)12-s + (−0.0627 − 0.998i)13-s + (0.809 − 0.587i)14-s + (−0.125 + 0.992i)15-s + (−0.929 − 0.368i)16-s + (−0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.329 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.329 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $-0.329 - 0.943i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ -0.329 - 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3386097516 - 0.4770703374i\)
\(L(\frac12)\) \(\approx\) \(0.3386097516 - 0.4770703374i\)
\(L(1)\) \(\approx\) \(0.6830833874 + 0.02938547658i\)
\(L(1)\) \(\approx\) \(0.6830833874 + 0.02938547658i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (-0.770 + 0.637i)T \)
3 \( 1 + (0.844 - 0.535i)T \)
5 \( 1 + (-0.637 + 0.770i)T \)
7 \( 1 + (-0.998 - 0.0627i)T \)
11 \( 1 + (0.904 + 0.425i)T \)
13 \( 1 + (-0.0627 - 0.998i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.929 + 0.368i)T \)
23 \( 1 + (-0.968 - 0.248i)T \)
29 \( 1 + (-0.998 + 0.0627i)T \)
31 \( 1 + (0.0627 - 0.998i)T \)
37 \( 1 + (0.535 - 0.844i)T \)
41 \( 1 + (-0.951 - 0.309i)T \)
43 \( 1 + (-0.728 + 0.684i)T \)
47 \( 1 + (-0.728 - 0.684i)T \)
53 \( 1 + (0.982 - 0.187i)T \)
59 \( 1 + (-0.368 + 0.929i)T \)
61 \( 1 + (0.982 + 0.187i)T \)
67 \( 1 + (-0.844 - 0.535i)T \)
71 \( 1 + (0.535 + 0.844i)T \)
73 \( 1 + (0.248 - 0.968i)T \)
79 \( 1 + (0.968 - 0.248i)T \)
83 \( 1 + (-0.248 - 0.968i)T \)
89 \( 1 + (0.368 + 0.929i)T \)
97 \( 1 + (-0.187 + 0.982i)T \)
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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

−29.95112774235319238102850359901, −28.623754228472018940323369104842, −27.89915539686483926159693344777, −26.9096223605294391395746758788, −26.058031268754121770081289278776, −25.1786621684503038507610190490, −23.93046934198140183903082380722, −22.133234926741968331451494814816, −21.44913277956027102499867808372, −20.15169371946849401046565175986, −19.53561408331375318103269792585, −18.92610220140377650486167464727, −16.92203164285255824127610123947, −16.35356205207301152546134482082, −15.23924089417114336401963854616, −13.55659187171242431425402599908, −12.52388162425584673335116096193, −11.32326880009109004230954583580, −9.91236930343672874933706045473, −8.98590823117991876756020274364, −8.309859808415618492705394706406, −6.74600526551484452100583684441, −4.23930948986499851331881122347, −3.49265704014731090482368340880, −1.77440561795706056816504916675, 0.28828211384869158179308660289, 2.33047201969241254598849170143, 3.821638919395119310137923981608, 6.23812339067925019909955517890, 7.07693930195992402453846959090, 8.0300132722667403960556393728, 9.30509961720987533192956642230, 10.27850695726340860420327860107, 11.87855214986776750096296051588, 13.34050863550180149848160015173, 14.66017415923040538109408406815, 15.25506497316353594088602564045, 16.46905283237554516016294847026, 17.94238745604937326161267344529, 18.75382057833965767830456104710, 19.70007987822673762563768941866, 20.22285446293654241128357145530, 22.45229610649176718089992398676, 23.19789868420505056268709600317, 24.50171634514737620564862525561, 25.39809788570798274594090549775, 26.08922782986560451262099765560, 27.03065493475159015453138416015, 27.97357352909708939284190873504, 29.58299745084750903807611207594

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