Properties

Degree 1
Conductor 101
Sign $-0.648 + 0.761i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(101\)
\( \varepsilon \)  =  $-0.648 + 0.761i$
motivic weight  =  \(0\)
character  :  $\chi_{101} (7, \cdot )$
Sato-Tate  :  $\mu(100)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 101,\ (1:\ ),\ -0.648 + 0.761i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.395253749 + 3.019945079i$
$L(\frac12,\chi)$  $\approx$  $1.395253749 + 3.019945079i$
$L(\chi,1)$  $\approx$  1.495504459 + 1.416424714i
$L(1,\chi)$  $\approx$  1.495504459 + 1.416424714i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

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