↑

Properties

Label 2-2100-21.5-c1-0-49
Degree $2$
Conductor $2100$
Sign $-0.992 + 0.122i$
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  + (0.417 − 1.68i)3-s + (−1.25 − 2.33i)7-s + (−2.65 − 1.40i)9-s + (4.63 − 2.67i)11-s + 4.13i·13-s + (−0.0773 − 0.134i)17-s + (−3.40 − 1.96i)19-s + (−4.44 + 1.13i)21-s + (−5.19 − 2.99i)23-s + (−3.46 + 3.86i)27-s − 10.3i·29-s + (6.70 − 3.86i)31-s + (−2.56 − 8.91i)33-s + (−5.24 + 9.07i)37-s + (6.95 + 1.72i)39-s + ⋯
L(s)  = 1  + (0.241 − 0.970i)3-s + (−0.473 − 0.880i)7-s + (−0.883 − 0.468i)9-s + (1.39 − 0.807i)11-s + 1.14i·13-s + (−0.0187 − 0.0325i)17-s + (−0.781 − 0.450i)19-s + (−0.968 + 0.247i)21-s + (−1.08 − 0.625i)23-s + (−0.667 + 0.744i)27-s − 1.91i·29-s + (1.20 − 0.694i)31-s + (−0.446 − 1.55i)33-s + (−0.861 + 1.49i)37-s + (1.11 + 0.276i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.992 + 0.122i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.992 + 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.227775764\)
\(L(\frac12)\) \(\approx\) \(1.227775764\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.417 + 1.68i)T \)
5 \( 1 \)
7 \( 1 + (1.25 + 2.33i)T \)
good11 \( 1 + (-4.63 + 2.67i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.13iT - 13T^{2} \)
17 \( 1 + (0.0773 + 0.134i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.40 + 1.96i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.19 + 2.99i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 10.3iT - 29T^{2} \)
31 \( 1 + (-6.70 + 3.86i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.24 - 9.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.33T + 41T^{2} \)
43 \( 1 - 1.78T + 43T^{2} \)
47 \( 1 + (-1.80 + 3.11i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.29 + 2.47i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.27 + 9.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.25 + 1.87i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.444 - 0.770i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 + (10.6 - 6.12i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.61 + 6.25i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.14T + 83T^{2} \)
89 \( 1 + (8.58 - 14.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.28iT - 97T^{2} \)
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

−8.553026528076091870009116133437, −8.069047152954373047182739486139, −6.91230639349990082818590387269, −6.53034272225425826344907247077, −6.01248100944194756924590335056, −4.36015233182483415764239747003, −3.84730581505927280657222104756, −2.65466405291634763844140442686, −1.55736088951155096235866116681, −0.41034325895651313980202021072, 1.75334500606074949230561179049, 2.93908081622173557494544954431, 3.70518706580168750855729996032, 4.55811069515210714770448581410, 5.53055641656815776880408204726, 6.13046858857455562431117942393, 7.13693910648359362695642591643, 8.167555190433399184803930774901, 8.957528549077162309342610929302, 9.288712218213624853348507181593

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