Properties

Label 2-2100-21.20-c1-0-40
Degree $2$
Conductor $2100$
Sign $-0.940 + 0.338i$
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  + (−0.586 − 1.62i)3-s − 2.64i·7-s + (−2.31 + 1.91i)9-s + 0.359i·11-s − 4.48i·13-s + 7.99·17-s + (−4.31 + 1.55i)21-s + (4.47 + 2.64i)27-s − 10.7i·29-s + (0.586 − 0.211i)33-s + (−7.31 + 2.63i)39-s − 12.4·47-s − 7.00·49-s + (−4.68 − 13.0i)51-s + (5.05 + 6.11i)63-s + ⋯
L(s)  = 1  + (−0.338 − 0.940i)3-s − 0.999i·7-s + (−0.770 + 0.637i)9-s + 0.108i·11-s − 1.24i·13-s + 1.93·17-s + (−0.940 + 0.338i)21-s + (0.860 + 0.509i)27-s − 1.99i·29-s + (0.102 − 0.0367i)33-s + (−1.17 + 0.421i)39-s − 1.81·47-s − 49-s + (−0.656 − 1.82i)51-s + (0.637 + 0.770i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.940 + 0.338i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.940 + 0.338i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.215666198\)
\(L(\frac12)\) \(\approx\) \(1.215666198\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.586 + 1.62i)T \)
5 \( 1 \)
7 \( 1 + 2.64iT \)
good11 \( 1 - 0.359iT - 11T^{2} \)
13 \( 1 + 4.48iT - 13T^{2} \)
17 \( 1 - 7.99T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 10.7iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + 10.5iT - 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 - 8.94T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 15.0iT - 97T^{2} \)
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

−8.388976366598264055956624067590, −7.78643275474152386658688584658, −7.42881838885498760724340548308, −6.37789718027639817311014160820, −5.71182055892951360189273835222, −4.89127065390460481878256829612, −3.66757602569181712572078832711, −2.79016608213254387138747017238, −1.41222132808456929696604379093, −0.48082497055179577479366881988, 1.52332908405175634207205167007, 2.95526484758851228212125263782, 3.63709164529683102791959750102, 4.78939100476911651659480966370, 5.37076155331754255036257902569, 6.12655684537884869497831239354, 6.97991471889603071628968370435, 8.126480505069569547677692510520, 8.843191742290513102195119005313, 9.472715662264211653674573600521

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