↑

Properties

Label 2-2400-120.59-c1-0-53
Degree $2$
Conductor $2400$
Sign $0.999 - 0.0387i$
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 + 0.242i)3-s + 3.08·7-s + (2.88 + 0.831i)9-s − 2.54i·11-s + 5.06·13-s − 0.214·17-s + 2.60·19-s + (5.29 + 0.749i)21-s + 4.47i·23-s + (4.74 + 2.12i)27-s − 7.86·29-s + 4.58i·31-s + (0.617 − 4.36i)33-s − 7.67·37-s + (8.68 + 1.22i)39-s + ⋯
L(s)  = 1  + (0.990 + 0.139i)3-s + 1.16·7-s + (0.960 + 0.277i)9-s − 0.767i·11-s + 1.40·13-s − 0.0519·17-s + 0.598·19-s + (1.15 + 0.163i)21-s + 0.933i·23-s + (0.912 + 0.408i)27-s − 1.46·29-s + 0.823i·31-s + (0.107 − 0.760i)33-s − 1.26·37-s + (1.39 + 0.196i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.999 - 0.0387i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ 0.999 - 0.0387i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.300685871\)
\(L(\frac12)\) \(\approx\) \(3.300685871\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.242i)T \)
5 \( 1 \)
good7 \( 1 - 3.08T + 7T^{2} \)
11 \( 1 + 2.54iT - 11T^{2} \)
13 \( 1 - 5.06T + 13T^{2} \)
17 \( 1 + 0.214T + 17T^{2} \)
19 \( 1 - 2.60T + 19T^{2} \)
23 \( 1 - 4.47iT - 23T^{2} \)
29 \( 1 + 7.86T + 29T^{2} \)
31 \( 1 - 4.58iT - 31T^{2} \)
37 \( 1 + 7.67T + 37T^{2} \)
41 \( 1 + 9.26iT - 41T^{2} \)
43 \( 1 + 11.4iT - 43T^{2} \)
47 \( 1 - 10.5iT - 47T^{2} \)
53 \( 1 + 9.51iT - 53T^{2} \)
59 \( 1 + 0.428iT - 59T^{2} \)
61 \( 1 + 1.11iT - 61T^{2} \)
67 \( 1 - 2.35iT - 67T^{2} \)
71 \( 1 - 6.12T + 71T^{2} \)
73 \( 1 + 12.0iT - 73T^{2} \)
79 \( 1 - 11.6iT - 79T^{2} \)
83 \( 1 + 2.29T + 83T^{2} \)
89 \( 1 - 12.4iT - 89T^{2} \)
97 \( 1 - 8.04iT - 97T^{2} \)
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

−8.830135458319857704336570251837, −8.311607110116632275300534384671, −7.63352660129566826286379447203, −6.87095423760059615291644261278, −5.65652919867645570195077527132, −5.05467070190060060699401554931, −3.76296887215587264347370678203, −3.48372215415570918273325205381, −2.05491696533085563880013716394, −1.26494230342593113088196601710, 1.30863560343420293993029842741, 2.02205352796560601277981280359, 3.18057943186363045914665093367, 4.11498152007837586864789756178, 4.77735334168201847016440959108, 5.86430363169022058193079882582, 6.84020973983969627414339570770, 7.62930123045156283058727889207, 8.194208506563315102798596104346, 8.818100932307814258628274695634

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