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Properties

Label 1-101-101.40-r1-0-0
Degree $1$
Conductor $101$
Sign $0.849 + 0.527i$
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  + (−0.125 + 0.992i)2-s + (−0.684 − 0.728i)3-s + (−0.968 − 0.248i)4-s + (−0.992 + 0.125i)5-s + (0.809 − 0.587i)6-s + (−0.904 + 0.425i)7-s + (0.368 − 0.929i)8-s + (−0.0627 + 0.998i)9-s − i·10-s + (−0.998 − 0.0627i)11-s + (0.481 + 0.876i)12-s + (0.425 − 0.904i)13-s + (−0.309 − 0.951i)14-s + (0.770 + 0.637i)15-s + (0.876 + 0.481i)16-s + (0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.125 + 0.992i)2-s + (−0.684 − 0.728i)3-s + (−0.968 − 0.248i)4-s + (−0.992 + 0.125i)5-s + (0.809 − 0.587i)6-s + (−0.904 + 0.425i)7-s + (0.368 − 0.929i)8-s + (−0.0627 + 0.998i)9-s − i·10-s + (−0.998 − 0.0627i)11-s + (0.481 + 0.876i)12-s + (0.425 − 0.904i)13-s + (−0.309 − 0.951i)14-s + (0.770 + 0.637i)15-s + (0.876 + 0.481i)16-s + (0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.849 + 0.527i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ 0.849 + 0.527i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5788159620 + 0.1649644623i\)
\(L(\frac12)\) \(\approx\) \(0.5788159620 + 0.1649644623i\)
\(L(1)\) \(\approx\) \(0.5334822590 + 0.1476242817i\)
\(L(1)\) \(\approx\) \(0.5334822590 + 0.1476242817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (-0.125 + 0.992i)T \)
3 \( 1 + (-0.684 - 0.728i)T \)
5 \( 1 + (-0.992 + 0.125i)T \)
7 \( 1 + (-0.904 + 0.425i)T \)
11 \( 1 + (-0.998 - 0.0627i)T \)
13 \( 1 + (0.425 - 0.904i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (0.876 - 0.481i)T \)
23 \( 1 + (0.187 + 0.982i)T \)
29 \( 1 + (-0.904 - 0.425i)T \)
31 \( 1 + (-0.425 - 0.904i)T \)
37 \( 1 + (0.728 + 0.684i)T \)
41 \( 1 + (0.587 + 0.809i)T \)
43 \( 1 + (-0.535 + 0.844i)T \)
47 \( 1 + (-0.535 - 0.844i)T \)
53 \( 1 + (0.248 + 0.968i)T \)
59 \( 1 + (0.481 - 0.876i)T \)
61 \( 1 + (0.248 - 0.968i)T \)
67 \( 1 + (0.684 - 0.728i)T \)
71 \( 1 + (0.728 - 0.684i)T \)
73 \( 1 + (-0.982 + 0.187i)T \)
79 \( 1 + (-0.187 + 0.982i)T \)
83 \( 1 + (0.982 + 0.187i)T \)
89 \( 1 + (-0.481 - 0.876i)T \)
97 \( 1 + (0.968 + 0.248i)T \)
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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

−29.08854856989269347542732491491, −28.718467273755684687210130607366, −27.61167170574278373233114346988, −26.72243471943444061466370614222, −26.05619707316537061583064092433, −23.73820931300900848310363407717, −23.06043821568655445673397298422, −22.327267880344522378125213378397, −20.96844633568377419781137466432, −20.33347836004320741294816096208, −19.03291396704851194100560896736, −18.19211855641454169766256434609, −16.49624680843360025474648262211, −16.12688674304992452521467625101, −14.44016185453143656321237446806, −12.89029148561747340778168115182, −11.9706428395837403607661410848, −10.95832961526691929146677146931, −10.02978565356827780219343756259, −8.90882315874295392182294461721, −7.294974331537675928285113271451, −5.36226362208958720994847123446, −4.10147072176905093601447087574, −3.192933280751549594424705418499, −0.64559761717705047500653466059, 0.59240285595852523903397819463, 3.32931121013904999420704800838, 5.20997189762412410466870223926, 6.12930313804932109474883522307, 7.48187476486951880299187517780, 8.093999102619634745519393124092, 9.84254282740935791786226607375, 11.26958684659496735481317781081, 12.70953792611633753189152546269, 13.324815106127012877722231995728, 15.133067849452494267383838543326, 15.89779154886458549046458288236, 16.81299816701260722282340261310, 18.214415703593375499174557313036, 18.76499932446050381676499686958, 19.82796394013595001134394245773, 21.929021592177607050548821882669, 22.95440908399282865898670522816, 23.42378493530796077047831668696, 24.44728106196316693173722335159, 25.528777250482953104461983627188, 26.44119832453327694011675958108, 27.85120610373059490160759544830, 28.32537420536743912790695116272, 29.73806685137387368478649293615

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