↑

Properties

Label 2-2700-15.2-c1-0-15
Degree $2$
Conductor $2700$
Sign $0.229 + 0.973i$
Analytic cond. $21.5596$
Root an. cond. $4.64323$
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  + (−2.44 − 2.44i)7-s + 6i·11-s + (−2.44 + 2.44i)13-s + (3.67 − 3.67i)17-s − i·19-s + (−3.67 − 3.67i)23-s + 6·29-s − 5·31-s + (4.89 + 4.89i)37-s − 6i·41-s + (7.34 − 7.34i)43-s + (7.34 − 7.34i)47-s + 4.99i·49-s + (−3.67 − 3.67i)53-s − 6·59-s + ⋯
L(s)  = 1  + (−0.925 − 0.925i)7-s + 1.80i·11-s + (−0.679 + 0.679i)13-s + (0.891 − 0.891i)17-s − 0.229i·19-s + (−0.766 − 0.766i)23-s + 1.11·29-s − 0.898·31-s + (0.805 + 0.805i)37-s − 0.937i·41-s + (1.12 − 1.12i)43-s + (1.07 − 1.07i)47-s + 0.714i·49-s + (−0.504 − 0.504i)53-s − 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(21.5596\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1/2),\ 0.229 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.142009096\)
\(L(\frac12)\) \(\approx\) \(1.142009096\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 + (2.44 + 2.44i)T + 7iT^{2} \)
11 \( 1 - 6iT - 11T^{2} \)
13 \( 1 + (2.44 - 2.44i)T - 13iT^{2} \)
17 \( 1 + (-3.67 + 3.67i)T - 17iT^{2} \)
19 \( 1 + iT - 19T^{2} \)
23 \( 1 + (3.67 + 3.67i)T + 23iT^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 + (-4.89 - 4.89i)T + 37iT^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 + (-7.34 + 7.34i)T - 43iT^{2} \)
47 \( 1 + (-7.34 + 7.34i)T - 47iT^{2} \)
53 \( 1 + (3.67 + 3.67i)T + 53iT^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 5T + 61T^{2} \)
67 \( 1 + (-4.89 - 4.89i)T + 67iT^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 + (2.44 - 2.44i)T - 73iT^{2} \)
79 \( 1 + 5iT - 79T^{2} \)
83 \( 1 + (11.0 + 11.0i)T + 83iT^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (4.89 + 4.89i)T + 97iT^{2} \)
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

−8.805610691110187051402669069189, −7.62095845096822707130402420280, −7.11631292344367795416335413163, −6.70606959439602395903047716670, −5.53369156097336284924197910077, −4.55089430301740172171803989914, −4.06120260136158691780239782767, −2.88154404996903000900411797638, −1.94306323267624226858222801353, −0.42925747731457629853951285944, 1.03448946570830639629881341384, 2.63471857107639093282827893908, 3.17837960413757202691485494361, 4.07646857149912752157073722965, 5.52152872795856968047648375588, 5.82325400006583864806974358005, 6.43882962063262631327147914004, 7.78455409937622461330147350571, 8.109201057659345242494400316887, 9.100910085683758442055392773951

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