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Properties

Label 1-101-101.15-r1-0-0
Degree $1$
Conductor $101$
Sign $0.222 + 0.975i$
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  + (0.904 − 0.425i)2-s + (0.481 + 0.876i)3-s + (0.637 − 0.770i)4-s + (−0.425 + 0.904i)5-s + (0.809 + 0.587i)6-s + (−0.684 + 0.728i)7-s + (0.248 − 0.968i)8-s + (−0.535 + 0.844i)9-s + i·10-s + (0.844 + 0.535i)11-s + (0.982 + 0.187i)12-s + (−0.728 + 0.684i)13-s + (−0.309 + 0.951i)14-s + (−0.998 + 0.0627i)15-s + (−0.187 − 0.982i)16-s + (0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (0.904 − 0.425i)2-s + (0.481 + 0.876i)3-s + (0.637 − 0.770i)4-s + (−0.425 + 0.904i)5-s + (0.809 + 0.587i)6-s + (−0.684 + 0.728i)7-s + (0.248 − 0.968i)8-s + (−0.535 + 0.844i)9-s + i·10-s + (0.844 + 0.535i)11-s + (0.982 + 0.187i)12-s + (−0.728 + 0.684i)13-s + (−0.309 + 0.951i)14-s + (−0.998 + 0.0627i)15-s + (−0.187 − 0.982i)16-s + (0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.222 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.222 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.222 + 0.975i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ 0.222 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.228535367 + 1.778073470i\)
\(L(\frac12)\) \(\approx\) \(2.228535367 + 1.778073470i\)
\(L(1)\) \(\approx\) \(1.759210893 + 0.5698041868i\)
\(L(1)\) \(\approx\) \(1.759210893 + 0.5698041868i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (0.904 - 0.425i)T \)
3 \( 1 + (0.481 + 0.876i)T \)
5 \( 1 + (-0.425 + 0.904i)T \)
7 \( 1 + (-0.684 + 0.728i)T \)
11 \( 1 + (0.844 + 0.535i)T \)
13 \( 1 + (-0.728 + 0.684i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.187 + 0.982i)T \)
23 \( 1 + (0.992 - 0.125i)T \)
29 \( 1 + (-0.684 - 0.728i)T \)
31 \( 1 + (0.728 + 0.684i)T \)
37 \( 1 + (0.876 + 0.481i)T \)
41 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 + (0.929 + 0.368i)T \)
47 \( 1 + (0.929 - 0.368i)T \)
53 \( 1 + (-0.770 + 0.637i)T \)
59 \( 1 + (0.982 - 0.187i)T \)
61 \( 1 + (-0.770 - 0.637i)T \)
67 \( 1 + (-0.481 + 0.876i)T \)
71 \( 1 + (0.876 - 0.481i)T \)
73 \( 1 + (-0.125 - 0.992i)T \)
79 \( 1 + (-0.992 - 0.125i)T \)
83 \( 1 + (0.125 - 0.992i)T \)
89 \( 1 + (-0.982 - 0.187i)T \)
97 \( 1 + (-0.637 + 0.770i)T \)
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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

−29.83493267859123098272933500747, −28.79088005793365649699504659625, −27.14598930742236220716845142191, −25.931559685470528743790693368020, −25.00657207958926224171508684552, −24.18286503876605983000526170530, −23.44632330335550872050561948183, −22.41200910592022484366466143281, −20.95875195961935709496503583387, −19.88229605299304904752176439887, −19.37316409620248288103264099296, −17.28871520040324560857051572281, −16.70547098727099125858073784750, −15.295474580034476002128260897198, −14.2131571822693774252671168735, −13.043676234566917304765630728943, −12.59990511679218483599016795218, −11.32464096349101608260480342789, −9.19397184661163302542163651797, −7.92943424144806102795705105444, −7.02695704731430709825402847833, −5.74914794342959337182184137028, −4.12798457334424874997565416566, −2.97535055663525170583178888169, −0.91641772686170193005340052930, 2.38201092341138310130477929305, 3.378403116852398987774734950830, 4.480253019327085016352025983, 5.99351235687147116666527907436, 7.318317868241039296277970751923, 9.33985317479010925252163586949, 10.146052269246992763264489820353, 11.471362823645664933619956525796, 12.36455259988845340648286507282, 14.08036185839787387741714367811, 14.73820661441871424505374097754, 15.5550158296065096115189588504, 16.68090420727135068567359921445, 18.931508627490711928817386192939, 19.35015563150456951764587713840, 20.631844629466327745588782212965, 21.6686986929377619635011646930, 22.48558328962824166270685687243, 23.06363833339370133932776516533, 24.86131251338332590275072518826, 25.55769403485135841185351614996, 26.89403477595449135250077633684, 27.81116947034866722985779331121, 28.95173959384375630567876573734, 30.08476600761446484701057691353

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