↑

Properties

Label 2-2100-105.62-c0-0-5
Degree $2$
Conductor $2100$
Sign $-0.0299 + 0.999i$
Analytic cond. $1.04803$
Root an. cond. $1.02373$
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  + (−0.258 + 0.965i)3-s + (−0.707 + 0.707i)7-s + (−0.866 − 0.499i)9-s − 1.73i·11-s + (−0.707 − 0.707i)13-s + (−1.22 − 1.22i)17-s + (−0.500 − 0.866i)21-s + (0.707 − 0.707i)27-s − 1.73·29-s + (1.67 + 0.448i)33-s + (0.866 − 0.500i)39-s + (1.22 + 1.22i)47-s − 1.00i·49-s + (1.49 − 0.866i)51-s + (0.965 − 0.258i)63-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)3-s + (−0.707 + 0.707i)7-s + (−0.866 − 0.499i)9-s − 1.73i·11-s + (−0.707 − 0.707i)13-s + (−1.22 − 1.22i)17-s + (−0.500 − 0.866i)21-s + (0.707 − 0.707i)27-s − 1.73·29-s + (1.67 + 0.448i)33-s + (0.866 − 0.500i)39-s + (1.22 + 1.22i)47-s − 1.00i·49-s + (1.49 − 0.866i)51-s + (0.965 − 0.258i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0299 + 0.999i$
Analytic conductor: \(1.04803\)
Root analytic conductor: \(1.02373\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (2057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :0),\ -0.0299 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3908062248\)
\(L(\frac12)\) \(\approx\) \(0.3908062248\)
\(L(1)\) 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 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 + 1.73iT - T^{2} \)
13 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
17 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + 1.73T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
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

−9.143604353614479549474948401126, −8.659102868117071935943441959170, −7.60296824848240014304343771188, −6.51951082459183972920723039447, −5.73270237183867089668558820625, −5.27921845411633366867182406037, −4.17418227821669131734218705865, −3.17305346256028909147446746792, −2.61652480915699039288634702198, −0.26404630221946519279447254916, 1.71276494134743300240203812512, 2.37329597507435368852888912578, 3.87457871573254250416683507654, 4.58440935401040042313365280920, 5.69626043129586567798778718291, 6.63449778600294310569600708940, 7.11599646064126455582388902905, 7.59779598528324878743823523163, 8.725776025793042739115247657107, 9.473761006526495151673043249086

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