↑

Properties

Label 1-99-99.97-r0-0-0
Degree $1$
Conductor $99$
Sign $0.996 + 0.0805i$
Analytic cond. $0.459754$
Root an. cond. $0.459754$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.104 + 0.994i)2-s + (−0.978 − 0.207i)4-s + (−0.104 − 0.994i)5-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + 10-s + (0.913 − 0.406i)13-s + (0.669 + 0.743i)14-s + (0.913 + 0.406i)16-s + (−0.809 + 0.587i)17-s + (0.309 − 0.951i)19-s + (−0.104 + 0.994i)20-s + (−0.5 + 0.866i)23-s + (−0.978 + 0.207i)25-s + (0.309 + 0.951i)26-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)2-s + (−0.978 − 0.207i)4-s + (−0.104 − 0.994i)5-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + 10-s + (0.913 − 0.406i)13-s + (0.669 + 0.743i)14-s + (0.913 + 0.406i)16-s + (−0.809 + 0.587i)17-s + (0.309 − 0.951i)19-s + (−0.104 + 0.994i)20-s + (−0.5 + 0.866i)23-s + (−0.978 + 0.207i)25-s + (0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.996 + 0.0805i$
Analytic conductor: \(0.459754\)
Root analytic conductor: \(0.459754\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (0:\ ),\ 0.996 + 0.0805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9000876955 + 0.03632302831i\)
\(L(\frac12)\) \(\approx\) \(0.9000876955 + 0.03632302831i\)
\(L(1)\) \(\approx\) \(0.9326140179 + 0.1463564050i\)
\(L(1)\) \(\approx\) \(0.9326140179 + 0.1463564050i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.104 + 0.994i)T \)
5 \( 1 + (-0.104 - 0.994i)T \)
7 \( 1 + (0.669 - 0.743i)T \)
13 \( 1 + (0.913 - 0.406i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.669 - 0.743i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (0.669 + 0.743i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.978 + 0.207i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (0.913 + 0.406i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.104 + 0.994i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.06657777425626106820395692294, −28.968151882812880866578448265157, −28.01702613090217201432153748228, −27.00751920106629364778804589698, −26.21043931357101295325204204614, −24.86690080191357199170912171088, −23.36339924382166886479686727175, −22.46726303798240850948596958923, −21.530923389491152332508388504227, −20.63094213571203901829453779956, −19.32896971630571627099815792947, −18.304709573516802875953104979187, −17.89677776963661484083627902366, −16.04317499033516544460585543715, −14.59571417866878612673216994822, −13.81883440031895827445280798739, −12.262653703767562776827791852926, −11.33502781085442950905892161532, −10.47476435017709062854099149871, −9.072898743612692606206981139984, −7.95642086190925589424916486480, −6.19175257642446132229255376357, −4.56265024187241615077915645341, −3.12702386928169929486265475354, −1.88516308008482645790602007985, 1.12551149452550444708906521196, 4.03394107966415499821133529502, 4.98557472121659342984986218648, 6.351622323988551152024551667278, 7.8489055154677419831836080946, 8.572989440571900295163913077861, 9.90117514716091914551294164898, 11.42497216608878937653394345275, 13.14543892078867765993869398836, 13.714453780117664234655074379374, 15.24679923319705219253089728239, 16.06907673274545223148650371141, 17.28795124971070072909286970621, 17.813179938653709583033106035340, 19.44984835075551354838877598476, 20.496966967256708991332833569707, 21.719789994229375024358460015466, 23.162052406300952590482917779682, 23.91409526120809162583113646073, 24.6596463839931946744745555570, 25.82016525313499948765678551090, 26.83427747385591820950681118751, 27.828807685351793252449358183964, 28.56983653387961815313626458244, 30.271933800128399283009736356094

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