Properties

Label 2-2100-21.20-c1-0-29
Degree $2$
Conductor $2100$
Sign $0.981 + 0.188i$
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  + 1.73i·3-s + (0.5 − 2.59i)7-s − 2.99·9-s + 1.73i·13-s − 5.19i·19-s + (4.5 + 0.866i)21-s − 5.19i·27-s − 1.73i·31-s + 10·37-s − 2.99·39-s + 13·43-s + (−6.5 − 2.59i)49-s + 9·57-s − 15.5i·61-s + (−1.49 + 7.79i)63-s + ⋯
L(s)  = 1  + 0.999i·3-s + (0.188 − 0.981i)7-s − 0.999·9-s + 0.480i·13-s − 1.19i·19-s + (0.981 + 0.188i)21-s − 0.999i·27-s − 0.311i·31-s + 1.64·37-s − 0.480·39-s + 1.98·43-s + (−0.928 − 0.371i)49-s + 1.19·57-s − 1.99i·61-s + (−0.188 + 0.981i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.981 + 0.188i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.981 + 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.579785580\)
\(L(\frac12)\) \(\approx\) \(1.579785580\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
5 \( 1 \)
7 \( 1 + (-0.5 + 2.59i)T \)
good11 \( 1 - 11T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 13T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 15.5iT - 61T^{2} \)
67 \( 1 - 11T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 13.8iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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

−9.392370219469317633615496673953, −8.357552991546846011215925798387, −7.60400890942188797442488002613, −6.71876485610259097606633427180, −5.85173560484428553557316107271, −4.77508339988735229769785415241, −4.31715088607608757116860871668, −3.41818524781695930131604683720, −2.34302864968474590573016515114, −0.65572117082124692658368154963, 1.08698661313618801790473441789, 2.23658816712640536133639029121, 2.97881154299357734548419582509, 4.23080757060767761464277536192, 5.58437134721207395232452429461, 5.80401584568793894548997609694, 6.79019970009611985461886590635, 7.71648291966824864121746920651, 8.212791670058767593650587445511, 8.959996768408671496646051101715

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