↑

Properties

Label 2-2400-20.7-c1-0-3
Degree $2$
Conductor $2400$
Sign $-0.991 - 0.130i$
Analytic cond. $19.1640$
Root an. cond. $4.37768$
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  + (0.707 + 0.707i)3-s + (−2.02 + 2.02i)7-s + 1.00i·9-s + 0.964i·11-s + (2.63 − 2.63i)13-s + (−2.14 − 2.14i)17-s − 4.76·19-s − 2.86·21-s + (−0.282 − 0.282i)23-s + (−0.707 + 0.707i)27-s + 7.32i·29-s + 8.42i·31-s + (−0.682 + 0.682i)33-s + (1.36 + 1.36i)37-s + 3.73·39-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.765 + 0.765i)7-s + 0.333i·9-s + 0.290i·11-s + (0.731 − 0.731i)13-s + (−0.520 − 0.520i)17-s − 1.09·19-s − 0.624·21-s + (−0.0589 − 0.0589i)23-s + (−0.136 + 0.136i)27-s + 1.36i·29-s + 1.51i·31-s + (−0.118 + 0.118i)33-s + (0.224 + 0.224i)37-s + 0.597·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $-0.991 - 0.130i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ -0.991 - 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7062332081\)
\(L(\frac12)\) \(\approx\) \(0.7062332081\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (2.02 - 2.02i)T - 7iT^{2} \)
11 \( 1 - 0.964iT - 11T^{2} \)
13 \( 1 + (-2.63 + 2.63i)T - 13iT^{2} \)
17 \( 1 + (2.14 + 2.14i)T + 17iT^{2} \)
19 \( 1 + 4.76T + 19T^{2} \)
23 \( 1 + (0.282 + 0.282i)T + 23iT^{2} \)
29 \( 1 - 7.32iT - 29T^{2} \)
31 \( 1 - 8.42iT - 31T^{2} \)
37 \( 1 + (-1.36 - 1.36i)T + 37iT^{2} \)
41 \( 1 + 8.05T + 41T^{2} \)
43 \( 1 + (3.30 + 3.30i)T + 43iT^{2} \)
47 \( 1 + (6.87 - 6.87i)T - 47iT^{2} \)
53 \( 1 + (-2.33 + 2.33i)T - 53iT^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + (1.03 - 1.03i)T - 67iT^{2} \)
71 \( 1 + 4.84iT - 71T^{2} \)
73 \( 1 + (-9.46 + 9.46i)T - 73iT^{2} \)
79 \( 1 + 1.53T + 79T^{2} \)
83 \( 1 + (8.94 + 8.94i)T + 83iT^{2} \)
89 \( 1 + 14.6iT - 89T^{2} \)
97 \( 1 + (12.9 + 12.9i)T + 97iT^{2} \)
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

−9.075692911774527517601295789663, −8.790012772188535212258138442363, −8.001273395835578725685563975833, −6.88618424854730008256113236119, −6.31395789729769780672516122189, −5.32921761478336798586435347675, −4.59937707438846817650121573658, −3.41813593921029777193724873081, −2.90069145972635517668449129075, −1.69964366618341123307539013129, 0.21140346300141796310424419735, 1.64255048184072407007492528554, 2.66251695133382490687076111757, 3.90035875956253628140548233122, 4.14994990411516912442653744476, 5.66017231576582676832195741829, 6.62655361432007238618799628949, 6.73282387504055379830164940130, 8.042072091712760332627982495037, 8.395807212334225622155433199777

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