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Properties

Label 1-101-101.7-r1-0-0
Degree $1$
Conductor $101$
Sign $-0.648 + 0.761i$
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  + (0.844 + 0.535i)2-s + (0.248 + 0.968i)3-s + (0.425 + 0.904i)4-s + (0.535 + 0.844i)5-s + (−0.309 + 0.951i)6-s + (0.368 − 0.929i)7-s + (−0.125 + 0.992i)8-s + (−0.876 + 0.481i)9-s + i·10-s + (0.481 + 0.876i)11-s + (−0.770 + 0.637i)12-s + (0.929 − 0.368i)13-s + (0.809 − 0.587i)14-s + (−0.684 + 0.728i)15-s + (−0.637 + 0.770i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.844 + 0.535i)2-s + (0.248 + 0.968i)3-s + (0.425 + 0.904i)4-s + (0.535 + 0.844i)5-s + (−0.309 + 0.951i)6-s + (0.368 − 0.929i)7-s + (−0.125 + 0.992i)8-s + (−0.876 + 0.481i)9-s + i·10-s + (0.481 + 0.876i)11-s + (−0.770 + 0.637i)12-s + (0.929 − 0.368i)13-s + (0.809 − 0.587i)14-s + (−0.684 + 0.728i)15-s + (−0.637 + 0.770i)16-s + (−0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $-0.648 + 0.761i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ -0.648 + 0.761i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.395253749 + 3.019945079i\)
\(L(\frac12)\) \(\approx\) \(1.395253749 + 3.019945079i\)
\(L(1)\) \(\approx\) \(1.495504459 + 1.416424714i\)
\(L(1)\) \(\approx\) \(1.495504459 + 1.416424714i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (0.844 + 0.535i)T \)
3 \( 1 + (0.248 + 0.968i)T \)
5 \( 1 + (0.535 + 0.844i)T \)
7 \( 1 + (0.368 - 0.929i)T \)
11 \( 1 + (0.481 + 0.876i)T \)
13 \( 1 + (0.929 - 0.368i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.637 - 0.770i)T \)
23 \( 1 + (-0.0627 - 0.998i)T \)
29 \( 1 + (0.368 + 0.929i)T \)
31 \( 1 + (-0.929 - 0.368i)T \)
37 \( 1 + (0.968 + 0.248i)T \)
41 \( 1 + (0.951 + 0.309i)T \)
43 \( 1 + (0.187 - 0.982i)T \)
47 \( 1 + (0.187 + 0.982i)T \)
53 \( 1 + (0.904 + 0.425i)T \)
59 \( 1 + (-0.770 - 0.637i)T \)
61 \( 1 + (0.904 - 0.425i)T \)
67 \( 1 + (-0.248 + 0.968i)T \)
71 \( 1 + (0.968 - 0.248i)T \)
73 \( 1 + (-0.998 + 0.0627i)T \)
79 \( 1 + (0.0627 - 0.998i)T \)
83 \( 1 + (0.998 + 0.0627i)T \)
89 \( 1 + (0.770 - 0.637i)T \)
97 \( 1 + (-0.425 - 0.904i)T \)
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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

−29.36927221190454815398241273388, −28.58147096351417889765464961167, −27.71791331199567000517687909212, −25.63345797588462827180635964282, −24.81437854660799902464054769668, −24.12505268547065890139956247274, −23.22061563163688041820712161082, −21.63541278409949857811920272749, −21.11947175733034119431343352674, −19.80330017387813592247766693615, −18.96227920384775014603247968947, −17.863681036634381769881898464760, −16.4017527484239564099956928146, −14.92571033947814747621739632823, −13.856112045579748851292695931, −13.03051394266438653477756769087, −12.106712278999316258240618322195, −11.13212931899743931354487053772, −9.23539136236694046592608229067, −8.298064101214907178850410652327, −6.197810790070443292990424489903, −5.70537515603739731487849640675, −3.87024775897472049996444669133, −2.17490267106082316051150942077, −1.23005952919800462814370974634, 2.52025501299764033742717367977, 3.85367754373153958982376648130, 4.84876959627632617203824858193, 6.34953919923478993805100062377, 7.46605982427120707973749968281, 9.02503620565988929247978942774, 10.52779649908798495071899878118, 11.288132067444079759656073494931, 13.15216558267846699449189218223, 14.20147102783031307059808368628, 14.80446344473031857651412654394, 15.93341039562838001760375317627, 17.06194632343840016529449855980, 17.98693533023433184222020185698, 20.10483222336096573234375535477, 20.72134716048091053632950265975, 21.88519548231764557515739882881, 22.64751051651655808300829520736, 23.476432588788365068690958242604, 25.098729492432104268982908362147, 25.83363323937022792064594049330, 26.61862302570951791488488282662, 27.677448344127996771216380979408, 29.2682394028816295529836906789, 30.43421413154615553368998414382

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