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Properties

Label 1-101-101.42-r1-0-0
Degree $1$
Conductor $101$
Sign $0.524 - 0.851i$
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  + (0.248 − 0.968i)2-s + (−0.998 − 0.0627i)3-s + (−0.876 − 0.481i)4-s + (0.968 − 0.248i)5-s + (−0.309 + 0.951i)6-s + (0.770 + 0.637i)7-s + (−0.684 + 0.728i)8-s + (0.992 + 0.125i)9-s − i·10-s + (−0.125 + 0.992i)11-s + (0.844 + 0.535i)12-s + (0.637 + 0.770i)13-s + (0.809 − 0.587i)14-s + (−0.982 + 0.187i)15-s + (0.535 + 0.844i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.248 − 0.968i)2-s + (−0.998 − 0.0627i)3-s + (−0.876 − 0.481i)4-s + (0.968 − 0.248i)5-s + (−0.309 + 0.951i)6-s + (0.770 + 0.637i)7-s + (−0.684 + 0.728i)8-s + (0.992 + 0.125i)9-s − i·10-s + (−0.125 + 0.992i)11-s + (0.844 + 0.535i)12-s + (0.637 + 0.770i)13-s + (0.809 − 0.587i)14-s + (−0.982 + 0.187i)15-s + (0.535 + 0.844i)16-s + (−0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.524 - 0.851i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ 0.524 - 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.466954230 - 0.8197235396i\)
\(L(\frac12)\) \(\approx\) \(1.466954230 - 0.8197235396i\)
\(L(1)\) \(\approx\) \(1.027370659 - 0.4730063885i\)
\(L(1)\) \(\approx\) \(1.027370659 - 0.4730063885i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (0.248 - 0.968i)T \)
3 \( 1 + (-0.998 - 0.0627i)T \)
5 \( 1 + (0.968 - 0.248i)T \)
7 \( 1 + (0.770 + 0.637i)T \)
11 \( 1 + (-0.125 + 0.992i)T \)
13 \( 1 + (0.637 + 0.770i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.535 - 0.844i)T \)
23 \( 1 + (0.929 - 0.368i)T \)
29 \( 1 + (0.770 - 0.637i)T \)
31 \( 1 + (-0.637 + 0.770i)T \)
37 \( 1 + (0.0627 + 0.998i)T \)
41 \( 1 + (-0.951 - 0.309i)T \)
43 \( 1 + (0.425 + 0.904i)T \)
47 \( 1 + (0.425 - 0.904i)T \)
53 \( 1 + (0.481 + 0.876i)T \)
59 \( 1 + (0.844 - 0.535i)T \)
61 \( 1 + (0.481 - 0.876i)T \)
67 \( 1 + (0.998 - 0.0627i)T \)
71 \( 1 + (0.0627 - 0.998i)T \)
73 \( 1 + (0.368 + 0.929i)T \)
79 \( 1 + (-0.929 - 0.368i)T \)
83 \( 1 + (-0.368 + 0.929i)T \)
89 \( 1 + (-0.844 - 0.535i)T \)
97 \( 1 + (0.876 + 0.481i)T \)
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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

−29.86868470863156074477878177714, −28.82340318846591059681963142881, −27.43598293299468867683793477409, −26.80195093072237790170426919776, −25.53985351171126385734847417533, −24.45391572394284760342092080889, −23.665962446244390828763507901987, −22.66406818307353324477404274693, −21.69235235145678219314018443498, −20.91528134133009588550256213352, −18.65436744348239598261419777137, −17.781481806637671115978104136664, −17.09551806056104758879613646263, −16.156846101431121540374324866057, −14.822404159645929184770711192717, −13.68631000849025370067973443040, −12.83077504166721338963749486478, −11.114593575954499783533011831348, −10.17406848224221623815451135242, −8.55334201470103344939060870644, −7.187266897738326587130799985519, −5.94266135303765149718357571235, −5.31381179958835213199694043728, −3.746493920094473796056984715431, −1.048812488519755823393434915348, 1.18066318487954007889844830957, 2.34020216647950864611903771314, 4.67844997408605147153045460229, 5.24308157081786779343686103094, 6.70311819316955491649377928688, 8.88756820423206638720149491616, 9.87892172616234484469289491997, 11.10425514078581953288886120939, 11.94027720266717317574405836560, 13.01406532482617629254019236029, 14.05536423447946034807190778737, 15.51312918222565612394514580858, 17.15365021742870807063639280588, 18.01048295767347284039685271595, 18.584342188182118157991031088693, 20.38239839783753252855669983936, 21.2493566769002804962237712996, 21.96815582022498884484906411033, 23.00697475152779633194120615001, 24.04779229012356490484267493901, 25.12403703706584303066468888260, 26.769865881949569489687175151420, 27.94022904105099360013050613665, 28.60043386032785332839010050556, 29.1866957089502096444823621092

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