↑

Properties

Label 2-2100-7.4-c1-0-17
Degree $2$
Conductor $2100$
Sign $-0.386 + 0.922i$
Analytic cond. $16.7685$
Root an. cond. $4.09494$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (−2 + 1.73i)7-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s + 2·13-s + (1 − 1.73i)17-s + (−2 − 3.46i)19-s + (0.499 + 2.59i)21-s + (−4 − 6.92i)23-s − 0.999·27-s + 4·29-s + (−1.5 + 2.59i)31-s + (0.999 + 1.73i)33-s + (−4.5 − 7.79i)37-s + (1 − 1.73i)39-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.755 + 0.654i)7-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s + 0.554·13-s + (0.242 − 0.420i)17-s + (−0.458 − 0.794i)19-s + (0.109 + 0.566i)21-s + (−0.834 − 1.44i)23-s − 0.192·27-s + 0.742·29-s + (−0.269 + 0.466i)31-s + (0.174 + 0.301i)33-s + (−0.739 − 1.28i)37-s + (0.160 − 0.277i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.146211421\)
\(L(\frac12)\) \(\approx\) \(1.146211421\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (2 - 1.73i)T \)
good11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.5 + 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.5 + 6.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 18T + 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.754206415704821322040433115907, −8.248553153495679047560846336136, −7.17591238316567771519067758180, −6.59621176562681413320486846632, −5.83711181549240987434063134822, −4.87133854066337045746307146690, −3.80089271795905957453570817546, −2.76298182539439411225729599734, −2.05496965683322539606227545092, −0.39438523929829473361109998470, 1.32468791285325361875660648977, 2.78664360504738746327124366774, 3.68192331937866003151033990781, 4.18367342422753220053866885927, 5.50023426234482004803686528520, 6.09569994569350684666552752670, 7.02192521188085583064093121710, 8.003108149277681927536420272119, 8.452743074243647658344693758958, 9.553141299068998996659063153651

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