Properties

Label 2-2240-5.4-c1-0-64
Degree $2$
Conductor $2240$
Sign $-0.994 - 0.100i$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
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  − 2.44i·3-s + (0.224 − 2.22i)5-s + i·7-s − 2.99·9-s + 4.89·11-s − 0.449i·13-s + (−5.44 − 0.550i)15-s + 2i·17-s − 6.44·19-s + 2.44·21-s − 6.89i·23-s + (−4.89 − i)25-s − 2.89·29-s + 0.898·31-s − 11.9i·33-s + ⋯
L(s)  = 1  − 1.41i·3-s + (0.100 − 0.994i)5-s + 0.377i·7-s − 0.999·9-s + 1.47·11-s − 0.124i·13-s + (−1.40 − 0.142i)15-s + 0.485i·17-s − 1.47·19-s + 0.534·21-s − 1.43i·23-s + (−0.979 − 0.200i)25-s − 0.538·29-s + 0.161·31-s − 2.08i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.994 - 0.100i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.994 - 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.396568384\)
\(L(\frac12)\) \(\approx\) \(1.396568384\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.224 + 2.22i)T \)
7 \( 1 - iT \)
good3 \( 1 + 2.44iT - 3T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 + 0.449iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 6.44T + 19T^{2} \)
23 \( 1 + 6.89iT - 23T^{2} \)
29 \( 1 + 2.89T + 29T^{2} \)
31 \( 1 - 0.898T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 8.89iT - 43T^{2} \)
47 \( 1 - 0.898iT - 47T^{2} \)
53 \( 1 + 1.10iT - 53T^{2} \)
59 \( 1 - 6.44T + 59T^{2} \)
61 \( 1 + 8.44T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 - 6.89iT - 73T^{2} \)
79 \( 1 + 2.89T + 79T^{2} \)
83 \( 1 + 2.44iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 3.79iT - 97T^{2} \)
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

−8.574624377861831963453777743852, −8.035918278607022359603332274925, −6.88223266194969221872064330762, −6.45471872466412330641976385089, −5.72239313020055373697278819409, −4.60843209416350743391751300481, −3.76883835224622892747157920678, −2.20644881370862982974725258072, −1.62048136249330780991156199022, −0.47118159281186383643883901959, 1.71883115554005322637905328162, 3.11624049135978475891754472030, 3.80178868217325949308372976117, 4.37484003347319707585405711598, 5.39343987891516992276119131784, 6.40484430119070004205901566417, 6.91212897690141551369137491111, 7.949082155655209248652476967243, 8.991679143777419681385334983288, 9.522250617569900122256215260286

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