↑

Properties

Label 1-2100-2100.11-r0-0-0
Degree $1$
Conductor $2100$
Sign $-0.999 + 0.0424i$
Analytic cond. $9.75235$
Root an. cond. $9.75235$
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  + (−0.978 + 0.207i)11-s + (0.309 + 0.951i)13-s + (−0.913 − 0.406i)17-s + (0.104 + 0.994i)19-s + (0.669 + 0.743i)23-s + (0.809 + 0.587i)29-s + (−0.913 − 0.406i)31-s + (−0.978 − 0.207i)37-s + (−0.309 − 0.951i)41-s − 43-s + (0.913 − 0.406i)47-s + (0.104 − 0.994i)53-s + (0.669 − 0.743i)59-s + (0.669 + 0.743i)61-s + (−0.913 − 0.406i)67-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)11-s + (0.309 + 0.951i)13-s + (−0.913 − 0.406i)17-s + (0.104 + 0.994i)19-s + (0.669 + 0.743i)23-s + (0.809 + 0.587i)29-s + (−0.913 − 0.406i)31-s + (−0.978 − 0.207i)37-s + (−0.309 − 0.951i)41-s − 43-s + (0.913 − 0.406i)47-s + (0.104 − 0.994i)53-s + (0.669 − 0.743i)59-s + (0.669 + 0.743i)61-s + (−0.913 − 0.406i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.999 + 0.0424i$
Analytic conductor: \(9.75235\)
Root analytic conductor: \(9.75235\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2100,\ (0:\ ),\ -0.999 + 0.0424i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005555993131 + 0.2618428401i\)
\(L(\frac12)\) \(\approx\) \(0.005555993131 + 0.2618428401i\)
\(L(1)\) \(\approx\) \(0.8081148382 + 0.1014382602i\)
\(L(1)\) \(\approx\) \(0.8081148382 + 0.1014382602i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (-0.913 - 0.406i)T \)
19 \( 1 + (0.104 + 0.994i)T \)
23 \( 1 + (0.669 + 0.743i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (0.913 - 0.406i)T \)
53 \( 1 + (0.104 - 0.994i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (-0.913 - 0.406i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (-0.669 - 0.743i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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

−19.512346074461963867861864349903, −18.723827073474129047343574153054, −17.95573911112934294037710544108, −17.494830155986749273054807557, −16.55771240363506536543157806414, −15.65874452970514717201936653480, −15.35080420257515463096350175219, −14.439797520892817220297779150663, −13.26579001300188292699764224120, −13.215710874540755204952089915654, −12.21349268205609008438470754438, −11.22385410372085356553760020689, −10.63040066741060282771477557306, −10.0360795846755187571707942693, −8.80904839569971208598969164217, −8.45784271650410825215418619357, −7.45431281586742218689338646324, −6.71350297581623517716345458463, −5.789449282060085820831122803, −5.02984833073436849105207013656, −4.25977069356559680009099066037, −3.05070767831050479900519932555, −2.574815837932830336891744773267, −1.29635725140440978656037038544, −0.086507530283525457397058258520, 1.48947814903200645781199037529, 2.25230966138432093096883069837, 3.29668499480470731772802367612, 4.13424971092865364726587082888, 5.0744821977076189504063797765, 5.712280583169418782612192363504, 6.86612892569592191021082060320, 7.28434683632384896522252927874, 8.4047358522333559723594422206, 8.95448766585244511324360388551, 9.89805576518395824407189911017, 10.59698949660940740304120886656, 11.38483301638864880460073666198, 12.08790148694822338668692671428, 12.98425200136861298310897312832, 13.5948354447786411927505975188, 14.32548370541619798767066598360, 15.19067455894202653637896933944, 15.90778608269634024652692798108, 16.45517836153670987611189248905, 17.37115471370867991943057874089, 18.096543116946028588373748648747, 18.71933147743042911053158898867, 19.38650577879610073965037019496, 20.32523619851356439741445752492

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