↑

Properties

Label 1-99-99.5-r1-0-0
Degree $1$
Conductor $99$
Sign $0.0805 + 0.996i$
Analytic cond. $10.6390$
Root an. cond. $10.6390$
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

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

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.0805 + 0.996i$
Analytic conductor: \(10.6390\)
Root analytic conductor: \(10.6390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (1:\ ),\ 0.0805 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.323065686 + 1.220423221i\)
\(L(\frac12)\) \(\approx\) \(1.323065686 + 1.220423221i\)
\(L(1)\) \(\approx\) \(1.065776547 + 0.5726799819i\)
\(L(1)\) \(\approx\) \(1.065776547 + 0.5726799819i\)

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

−29.834098044975568021064839871010, −28.56116989538361431457545709890, −27.43548533924638566447609519056, −26.68502555010572475599938010387, −25.637047457217972420804087992950, −23.90778863974263965598550574838, −22.99241583184029861193538695888, −22.15072316720653318266147568707, −20.93844030938481402659218585728, −20.215084012382348070939819647919, −18.82617358446832387763572666982, −18.14898731864724016956338216098, −17.0460727048521133586772151854, −15.184491347079391852633986407654, −14.095436694020189685996837725195, −13.34277692362880730814174832369, −11.72036297992842684331461093613, −10.86082853555645633715472327886, −10.04219283843087799042821252661, −8.49322878948863465373561065397, −7.07519175909121080112329368954, −5.347727107227446844293814107698, −3.87299905924880373818124037613, −2.67768686208666644446961854081, −0.96932535101209243553252746406, 1.3475337489348458251091627243, 3.86776509698060276246450437913, 5.211878790999980506062178506926, 6.03464953642237590756842339201, 7.85192781808624429255065453511, 8.61750443857073624759304810054, 9.71141732328793629233019435420, 11.70134585388793789174194785407, 12.809730293884851694386119921574, 13.91506941294883193135192714421, 15.06875139300860256356573337507, 16.11026188781268862759181253401, 17.02069469421266115734866285391, 18.08071096772885978075352131946, 19.13639254969798974264679220766, 20.91824367389411980394678579695, 21.495263245086381732669957072706, 23.042565271343559319020083806533, 23.87928373315259553699075139110, 24.874444461967917763174926870827, 25.46757976544743925250604127883, 26.85296125615109919698474168564, 27.90939253680868046978692893722, 28.50448589314261886546839868885, 30.275356090788808408326042105234

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