↑

Properties

Label 2-2400-24.11-c1-0-17
Degree $2$
Conductor $2400$
Sign $0.721 - 0.691i$
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  + (−1.71 + 0.218i)3-s − 3.64i·7-s + (2.90 − 0.750i)9-s + 5.07i·11-s − 1.70i·13-s + 4.08i·17-s + 1.26·19-s + (0.796 + 6.26i)21-s − 4.70·23-s + (−4.82 + 1.92i)27-s − 1.06·29-s − 4.86i·31-s + (−1.10 − 8.71i)33-s + 7.56i·37-s + (0.372 + 2.93i)39-s + ⋯
L(s)  = 1  + (−0.992 + 0.126i)3-s − 1.37i·7-s + (0.968 − 0.250i)9-s + 1.52i·11-s − 0.473i·13-s + 0.989i·17-s + 0.290·19-s + (0.173 + 1.36i)21-s − 0.980·23-s + (−0.928 + 0.370i)27-s − 0.197·29-s − 0.874i·31-s + (−0.192 − 1.51i)33-s + 1.24i·37-s + (0.0596 + 0.469i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.721 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.721 - 0.691i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (2351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ 0.721 - 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.039641579\)
\(L(\frac12)\) \(\approx\) \(1.039641579\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 - 0.218i)T \)
5 \( 1 \)
good7 \( 1 + 3.64iT - 7T^{2} \)
11 \( 1 - 5.07iT - 11T^{2} \)
13 \( 1 + 1.70iT - 13T^{2} \)
17 \( 1 - 4.08iT - 17T^{2} \)
19 \( 1 - 1.26T + 19T^{2} \)
23 \( 1 + 4.70T + 23T^{2} \)
29 \( 1 + 1.06T + 29T^{2} \)
31 \( 1 + 4.86iT - 31T^{2} \)
37 \( 1 - 7.56iT - 37T^{2} \)
41 \( 1 - 1.50iT - 41T^{2} \)
43 \( 1 - 3.43T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + 8.87T + 53T^{2} \)
59 \( 1 + 0.788iT - 59T^{2} \)
61 \( 1 - 0.627iT - 61T^{2} \)
67 \( 1 - 4.18T + 67T^{2} \)
71 \( 1 - 6.21T + 71T^{2} \)
73 \( 1 - 4.21T + 73T^{2} \)
79 \( 1 - 0.992iT - 79T^{2} \)
83 \( 1 + 7.72iT - 83T^{2} \)
89 \( 1 - 11.5iT - 89T^{2} \)
97 \( 1 - 7.40T + 97T^{2} \)
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

−9.355445056238347390132160374995, −7.920256090151785563990600857516, −7.52217898298729600893118158790, −6.70096965848564435168184924653, −6.03301477996843368380467416309, −4.99808720018079421480703972305, −4.28880265369294055987176358458, −3.70775817981484397110378991328, −2.01672745709468504300659006041, −0.920913559715668822368879691438, 0.52246117368257846605160147766, 1.95164468566419133614109045669, 3.00644065823070656414116060527, 4.13238631802817383962671812342, 5.25913794781711653889615265962, 5.69230976801532469412606709707, 6.32011039661978846561668507082, 7.22565073143168446397820077316, 8.129248695616458684662857282847, 8.982220628112069786384359103705

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