↑

Properties

Label 2-2100-105.104-c1-0-35
Degree $2$
Conductor $2100$
Sign $-0.775 + 0.631i$
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  + (−0.618 − 1.61i)3-s + (2.61 + 0.381i)7-s + (−2.23 + 2.00i)9-s − 0.763i·11-s − 1.23·13-s + 4.47i·17-s − 7.23i·19-s + (−1.00 − 4.47i)21-s − 7.70·23-s + (4.61 + 2.38i)27-s − 4i·29-s − 3.23i·31-s + (−1.23 + 0.472i)33-s − 4.47i·37-s + (0.763 + 2.00i)39-s + ⋯
L(s)  = 1  + (−0.356 − 0.934i)3-s + (0.989 + 0.144i)7-s + (−0.745 + 0.666i)9-s − 0.230i·11-s − 0.342·13-s + 1.08i·17-s − 1.66i·19-s + (−0.218 − 0.975i)21-s − 1.60·23-s + (0.888 + 0.458i)27-s − 0.742i·29-s − 0.581i·31-s + (−0.215 + 0.0821i)33-s − 0.735i·37-s + (0.122 + 0.320i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.775 + 0.631i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.775 + 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.095234539\)
\(L(\frac12)\) \(\approx\) \(1.095234539\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.618 + 1.61i)T \)
5 \( 1 \)
7 \( 1 + (-2.61 - 0.381i)T \)
good11 \( 1 + 0.763iT - 11T^{2} \)
13 \( 1 + 1.23T + 13T^{2} \)
17 \( 1 - 4.47iT - 17T^{2} \)
19 \( 1 + 7.23iT - 19T^{2} \)
23 \( 1 + 7.70T + 23T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + 3.23iT - 31T^{2} \)
37 \( 1 + 4.47iT - 37T^{2} \)
41 \( 1 - 3.52T + 41T^{2} \)
43 \( 1 + 7.23iT - 43T^{2} \)
47 \( 1 + 3.23iT - 47T^{2} \)
53 \( 1 + 9.23T + 53T^{2} \)
59 \( 1 - 8.94T + 59T^{2} \)
61 \( 1 + 4.94iT - 61T^{2} \)
67 \( 1 + 9.70iT - 67T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 - 6.76T + 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 - 7.23iT - 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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

−8.529956873877314426534117945180, −8.000160373567353142452617498428, −7.33289504447586852257859031582, −6.41178603785979315643624976777, −5.72276864661048780414186123983, −4.90599165225926629902654198602, −3.95482620203923395606448534260, −2.47017194323441362000373258579, −1.81344595940210838506306642021, −0.40544156190771586883329953798, 1.40689975505047826845935252090, 2.74363081853808379896300640461, 3.88843138427396004700541952098, 4.55833648402376316353041337322, 5.33752129310957774086546495133, 6.02777147315538679873141166335, 7.11948284781107791398399911284, 8.011772442880686728556916841177, 8.579487565114659045944995154706, 9.687099742338552075275405290323

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