↑

Properties

Label 2-2100-35.4-c1-0-19
Degree $2$
Conductor $2100$
Sign $0.788 + 0.615i$
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  + (0.866 + 0.5i)3-s + (2.09 + 1.62i)7-s + (0.499 + 0.866i)9-s + (2.12 − 3.67i)11-s − 3.24i·13-s + (−3.67 − 2.12i)17-s + (−3.5 − 6.06i)19-s + (0.999 + 2.44i)21-s + (3.67 − 2.12i)23-s + 0.999i·27-s + 1.75·29-s + (4.74 − 8.21i)31-s + (3.67 − 2.12i)33-s + (2.80 − 1.62i)37-s + (1.62 − 2.80i)39-s + ⋯
L(s)  = 1  + (0.499 + 0.288i)3-s + (0.790 + 0.612i)7-s + (0.166 + 0.288i)9-s + (0.639 − 1.10i)11-s − 0.899i·13-s + (−0.891 − 0.514i)17-s + (−0.802 − 1.39i)19-s + (0.218 + 0.534i)21-s + (0.766 − 0.442i)23-s + 0.192i·27-s + 0.326·29-s + (0.851 − 1.47i)31-s + (0.639 − 0.369i)33-s + (0.461 − 0.266i)37-s + (0.259 − 0.449i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.788 + 0.615i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.788 + 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.251433423\)
\(L(\frac12)\) \(\approx\) \(2.251433423\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-2.09 - 1.62i)T \)
good11 \( 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.24iT - 13T^{2} \)
17 \( 1 + (3.67 + 2.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.67 + 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.75T + 29T^{2} \)
31 \( 1 + (-4.74 + 8.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.80 + 1.62i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + 3.24iT - 43T^{2} \)
47 \( 1 + (5.19 - 3i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.34 - 4.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.12 - 8.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.24 + 3.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.54 - 2.62i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (-8.00 - 4.62i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 + (5.12 + 8.87i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.485iT - 97T^{2} \)
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

−8.705657983179153261784353616697, −8.619831245832933634290174512415, −7.62961838192802663211376129569, −6.64871406558755304009996949728, −5.83245555105698813496493370354, −4.87886069119657525836654973162, −4.21941602544691223617504858928, −2.96322776930228144192367428412, −2.35495714475570489550628458882, −0.77795509260581961541259531352, 1.47508839883264783770891687683, 1.99237552250492068938053354448, 3.48259487610441857146031951067, 4.34286884759447799192582552849, 4.88267690386088910378404687482, 6.39335599295561276844067170216, 6.81607151018801189145417291411, 7.66819345808623863928638769307, 8.428817137054532193017815688672, 9.021950019854646103344055650218

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