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Properties

Label 1-101-101.54-r0-0-0
Degree $1$
Conductor $101$
Sign $0.892 + 0.450i$
Analytic cond. $0.469042$
Root an. cond. $0.469042$
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  + (0.876 + 0.481i)2-s + (−0.992 − 0.125i)3-s + (0.535 + 0.844i)4-s + (0.876 − 0.481i)5-s + (−0.809 − 0.587i)6-s + (−0.187 − 0.982i)7-s + (0.0627 + 0.998i)8-s + (0.968 + 0.248i)9-s + 10-s + (0.968 + 0.248i)11-s + (−0.425 − 0.904i)12-s + (−0.187 + 0.982i)13-s + (0.309 − 0.951i)14-s + (−0.929 + 0.368i)15-s + (−0.425 + 0.904i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (0.876 + 0.481i)2-s + (−0.992 − 0.125i)3-s + (0.535 + 0.844i)4-s + (0.876 − 0.481i)5-s + (−0.809 − 0.587i)6-s + (−0.187 − 0.982i)7-s + (0.0627 + 0.998i)8-s + (0.968 + 0.248i)9-s + 10-s + (0.968 + 0.248i)11-s + (−0.425 − 0.904i)12-s + (−0.187 + 0.982i)13-s + (0.309 − 0.951i)14-s + (−0.929 + 0.368i)15-s + (−0.425 + 0.904i)16-s + (−0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.892 + 0.450i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ 0.892 + 0.450i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.362834455 + 0.3244427354i\)
\(L(\frac12)\) \(\approx\) \(1.362834455 + 0.3244427354i\)
\(L(1)\) \(\approx\) \(1.368553150 + 0.2572815007i\)
\(L(1)\) \(\approx\) \(1.368553150 + 0.2572815007i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (0.876 + 0.481i)T \)
3 \( 1 + (-0.992 - 0.125i)T \)
5 \( 1 + (0.876 - 0.481i)T \)
7 \( 1 + (-0.187 - 0.982i)T \)
11 \( 1 + (0.968 + 0.248i)T \)
13 \( 1 + (-0.187 + 0.982i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.425 - 0.904i)T \)
23 \( 1 + (0.728 - 0.684i)T \)
29 \( 1 + (-0.187 + 0.982i)T \)
31 \( 1 + (-0.187 - 0.982i)T \)
37 \( 1 + (-0.992 + 0.125i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.637 + 0.770i)T \)
47 \( 1 + (-0.637 - 0.770i)T \)
53 \( 1 + (0.535 - 0.844i)T \)
59 \( 1 + (-0.425 + 0.904i)T \)
61 \( 1 + (0.535 + 0.844i)T \)
67 \( 1 + (-0.992 + 0.125i)T \)
71 \( 1 + (-0.992 - 0.125i)T \)
73 \( 1 + (0.728 - 0.684i)T \)
79 \( 1 + (0.728 + 0.684i)T \)
83 \( 1 + (0.728 + 0.684i)T \)
89 \( 1 + (-0.425 - 0.904i)T \)
97 \( 1 + (0.535 + 0.844i)T \)
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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

−29.62447797798534942270148347469, −29.120378803981150473658432349648, −28.065563936410000141893089969825, −27.11216820054232873417008957448, −25.07381700117208500373905385886, −24.77510432630404521498672745366, −23.14523691070353672861153019527, −22.33360870601407319613391445109, −21.822426564255962412740608807513, −20.83062210544759157471906621076, −19.24605162366280244808791779583, −18.244517453012724039118248754583, −17.138217347511952048071034366415, −15.69720502652266110082949945985, −14.798805054264373438262363280104, −13.44804623399116806521555100445, −12.3942202079175567977615907179, −11.4392694099408447172780771763, −10.37260201248952164999614950880, −9.336453892327474608871731887784, −6.766527168632799065949512103228, −5.92936702235869557055345348418, −5.04689370045343902014065292540, −3.29747952480730430376622349802, −1.74753273103859450290890088433, 1.77310248968161559479277105650, 4.127403214309646045007036395125, 4.97867594209606735000813107406, 6.50550005772333221477025150490, 6.905485115714378812774675377429, 8.966398049897035235617408780466, 10.540533382660345910101349293351, 11.69201796792887742407217783090, 12.90365183222078634333959217099, 13.59223374890667296576904399464, 14.88529652393483681740373986845, 16.5319684503978930903320405870, 16.90619009410587522562815126555, 17.76855297872921352454475773884, 19.668458215997057911856950734772, 20.94172297058347350939577099288, 21.92729569877469112159145413513, 22.63370436304538705692458344243, 23.922467123426802432524560602313, 24.33208312137236230405271595857, 25.65666121632593190523159355885, 26.6899357240987607870122204583, 28.191660542741808200135008875768, 29.24524246172868405899378580181, 29.85605000906873997307826774976

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