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Properties

Label 1-101-101.81-r0-0-0
Degree $1$
Conductor $101$
Sign $0.675 - 0.737i$
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  + (0.0627 − 0.998i)2-s + (−0.929 + 0.368i)3-s + (−0.992 − 0.125i)4-s + (0.0627 + 0.998i)5-s + (0.309 + 0.951i)6-s + (0.535 − 0.844i)7-s + (−0.187 + 0.982i)8-s + (0.728 − 0.684i)9-s + 10-s + (0.728 − 0.684i)11-s + (0.968 − 0.248i)12-s + (0.535 + 0.844i)13-s + (−0.809 − 0.587i)14-s + (−0.425 − 0.904i)15-s + (0.968 + 0.248i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.0627 − 0.998i)2-s + (−0.929 + 0.368i)3-s + (−0.992 − 0.125i)4-s + (0.0627 + 0.998i)5-s + (0.309 + 0.951i)6-s + (0.535 − 0.844i)7-s + (−0.187 + 0.982i)8-s + (0.728 − 0.684i)9-s + 10-s + (0.728 − 0.684i)11-s + (0.968 − 0.248i)12-s + (0.535 + 0.844i)13-s + (−0.809 − 0.587i)14-s + (−0.425 − 0.904i)15-s + (0.968 + 0.248i)16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.675 - 0.737i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ 0.675 - 0.737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7347884619 - 0.3232399395i\)
\(L(\frac12)\) \(\approx\) \(0.7347884619 - 0.3232399395i\)
\(L(1)\) \(\approx\) \(0.8037781202 - 0.2716214714i\)
\(L(1)\) \(\approx\) \(0.8037781202 - 0.2716214714i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (0.0627 - 0.998i)T \)
3 \( 1 + (-0.929 + 0.368i)T \)
5 \( 1 + (0.0627 + 0.998i)T \)
7 \( 1 + (0.535 - 0.844i)T \)
11 \( 1 + (0.728 - 0.684i)T \)
13 \( 1 + (0.535 + 0.844i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.968 - 0.248i)T \)
23 \( 1 + (-0.637 + 0.770i)T \)
29 \( 1 + (0.535 + 0.844i)T \)
31 \( 1 + (0.535 - 0.844i)T \)
37 \( 1 + (-0.929 - 0.368i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.876 - 0.481i)T \)
47 \( 1 + (0.876 + 0.481i)T \)
53 \( 1 + (-0.992 + 0.125i)T \)
59 \( 1 + (0.968 + 0.248i)T \)
61 \( 1 + (-0.992 - 0.125i)T \)
67 \( 1 + (-0.929 - 0.368i)T \)
71 \( 1 + (-0.929 + 0.368i)T \)
73 \( 1 + (-0.637 + 0.770i)T \)
79 \( 1 + (-0.637 - 0.770i)T \)
83 \( 1 + (-0.637 - 0.770i)T \)
89 \( 1 + (0.968 - 0.248i)T \)
97 \( 1 + (-0.992 - 0.125i)T \)
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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

−30.27343973763352543128425496665, −28.454632229668589426627616139761, −28.08441981491639731923799875292, −27.18524910079403849359965756943, −25.42492489136153434387809471959, −24.72030606025678649750757214781, −24.04430022166557894119316210087, −22.91660796845392349571117234994, −22.07199062226908083974200224100, −20.84364659059615786558502999180, −19.16032066241249646236484707888, −17.83485452869523738255904183809, −17.42401678253774624241813324252, −16.21405129752682167576905447441, −15.37507671097523125933486883530, −13.921193509987513093030284033477, −12.5614339392612833711559473621, −12.06215568365622981907236071225, −10.134243976758018315501940770231, −8.71388528338066284868171938986, −7.75755404586758557132917700305, −6.18238514033549218559171642191, −5.39738920193422626378997434817, −4.31546751353726768454047748705, −1.324414492063822793889243049606, 1.262234746627459748979411752308, 3.351191623545819969801897403253, 4.402276965274900571731771141286, 5.8878512840965547235326739522, 7.31310667878188613484576633173, 9.285902483424502254050573829647, 10.35257346316791924846674552322, 11.33642301402720694028410823809, 11.762709477691661446567305616768, 13.70365874909463055594185762504, 14.311030083956813707776902991531, 16.05331084055071289187404439591, 17.34941873718281187229256328703, 18.159234581746710060302385429390, 19.15084758858246863215700667218, 20.52304229448750768508026637262, 21.53696350877975972843334913028, 22.32543454131804244025735416456, 23.1879545391598144478203803477, 24.092663962273036703714417567966, 26.20734144805876291504399631830, 27.00066842481786577054868131055, 27.65582971525978959895908796705, 28.95144704377632754787212853228, 29.75848022120262217159745673114

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