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Properties

Label 1-101-101.45-r0-0-0
Degree $1$
Conductor $101$
Sign $-0.774 - 0.633i$
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  + (−0.728 − 0.684i)2-s + (0.187 − 0.982i)3-s + (0.0627 + 0.998i)4-s + (0.728 − 0.684i)5-s + (−0.809 + 0.587i)6-s + (−0.876 − 0.481i)7-s + (0.637 − 0.770i)8-s + (−0.929 − 0.368i)9-s − 10-s + (0.929 + 0.368i)11-s + (0.992 + 0.125i)12-s + (0.876 − 0.481i)13-s + (0.309 + 0.951i)14-s + (−0.535 − 0.844i)15-s + (−0.992 + 0.125i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (−0.728 − 0.684i)2-s + (0.187 − 0.982i)3-s + (0.0627 + 0.998i)4-s + (0.728 − 0.684i)5-s + (−0.809 + 0.587i)6-s + (−0.876 − 0.481i)7-s + (0.637 − 0.770i)8-s + (−0.929 − 0.368i)9-s − 10-s + (0.929 + 0.368i)11-s + (0.992 + 0.125i)12-s + (0.876 − 0.481i)13-s + (0.309 + 0.951i)14-s + (−0.535 − 0.844i)15-s + (−0.992 + 0.125i)16-s + (−0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $-0.774 - 0.633i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ -0.774 - 0.633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2522204522 - 0.7067767798i\)
\(L(\frac12)\) \(\approx\) \(0.2522204522 - 0.7067767798i\)
\(L(1)\) \(\approx\) \(0.5704824094 - 0.5621148750i\)
\(L(1)\) \(\approx\) \(0.5704824094 - 0.5621148750i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (-0.728 - 0.684i)T \)
3 \( 1 + (0.187 - 0.982i)T \)
5 \( 1 + (0.728 - 0.684i)T \)
7 \( 1 + (-0.876 - 0.481i)T \)
11 \( 1 + (0.929 + 0.368i)T \)
13 \( 1 + (0.876 - 0.481i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.992 - 0.125i)T \)
23 \( 1 + (-0.425 + 0.904i)T \)
29 \( 1 + (-0.876 + 0.481i)T \)
31 \( 1 + (0.876 + 0.481i)T \)
37 \( 1 + (-0.187 - 0.982i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + (0.968 + 0.248i)T \)
47 \( 1 + (0.968 - 0.248i)T \)
53 \( 1 + (-0.0627 + 0.998i)T \)
59 \( 1 + (0.992 - 0.125i)T \)
61 \( 1 + (-0.0627 - 0.998i)T \)
67 \( 1 + (0.187 + 0.982i)T \)
71 \( 1 + (-0.187 + 0.982i)T \)
73 \( 1 + (0.425 - 0.904i)T \)
79 \( 1 + (-0.425 - 0.904i)T \)
83 \( 1 + (0.425 + 0.904i)T \)
89 \( 1 + (0.992 + 0.125i)T \)
97 \( 1 + (0.0627 + 0.998i)T \)
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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

−30.16903710848590107914505215137, −28.81285075187104360274937273742, −28.17201284666984939233815354162, −26.98954269435625936734970447673, −25.98714453249027679596760523328, −25.66965537787793579196563384371, −24.446634345803961636899510595997, −22.78256417532442468615833097099, −22.142493126178357642378107430300, −20.91588224148280160373464860364, −19.49264762929698350506302347821, −18.7313078909363528367827434662, −17.33721625925509332935706903969, −16.50596594901029626755191331176, −15.42896963938833984554772541384, −14.57453002368248401282112040519, −13.50701189040094432214283836356, −11.26699449272150773621782215347, −10.29997914821892880139295662506, −9.30451516724228669530307528774, −8.553111622657179153972773664212, −6.44236703583909980518208821274, −6.007248731476847637198140864926, −4.06706171588964116992246162714, −2.31188851195097211472098718494, 0.994210261658389114077150649743, 2.28505913109836704029744293374, 3.87400280231762118455240544636, 6.15156036587063785366843742235, 7.24045774608853299510775247891, 8.726586015773922406339119574284, 9.42398156384897144116394113967, 10.87621913166798342996682166845, 12.28353294763436928258229034099, 13.06395195531867589149024962516, 13.858371339597103633426977131288, 16.006895208002805254878627227299, 17.25014294867748758944478220377, 17.759129321666436964544588866397, 19.095181341268338475442713979307, 19.94295123169437210525386471094, 20.65159434067626140043817484029, 22.09852138366460746512680418178, 23.196890484918805059111805915775, 24.72228129479959331764645295641, 25.45919187619804069384661724366, 26.17300017538614443451320150659, 27.79122354256310710841468402724, 28.57401612866525709197460449447, 29.67535996227848157368210488185

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