Properties

Label 2-2240-5.4-c1-0-49
Degree $2$
Conductor $2240$
Sign $-0.553 + 0.833i$
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  − 1.83i·3-s + (−1.86 − 1.23i)5-s + i·7-s − 0.359·9-s + 4.40·11-s − 3.20i·13-s + (−2.26 + 3.41i)15-s + 1.14i·17-s + 1.72·19-s + 1.83·21-s − 3.25i·23-s + (1.94 + 4.60i)25-s − 4.83i·27-s + 4.18·29-s − 1.36·31-s + ⋯
L(s)  = 1  − 1.05i·3-s + (−0.833 − 0.553i)5-s + 0.377i·7-s − 0.119·9-s + 1.32·11-s − 0.889i·13-s + (−0.585 + 0.881i)15-s + 0.278i·17-s + 0.395·19-s + 0.399·21-s − 0.678i·23-s + (0.388 + 0.921i)25-s − 0.931i·27-s + 0.777·29-s − 0.245·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.553 + 0.833i$
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: \((2,\ 2240,\ (\ :1/2),\ -0.553 + 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.595929850\)
\(L(\frac12)\) \(\approx\) \(1.595929850\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 + 1.23i)T \)
7 \( 1 - iT \)
good3 \( 1 + 1.83iT - 3T^{2} \)
11 \( 1 - 4.40T + 11T^{2} \)
13 \( 1 + 3.20iT - 13T^{2} \)
17 \( 1 - 1.14iT - 17T^{2} \)
19 \( 1 - 1.72T + 19T^{2} \)
23 \( 1 + 3.25iT - 23T^{2} \)
29 \( 1 - 4.18T + 29T^{2} \)
31 \( 1 + 1.36T + 31T^{2} \)
37 \( 1 + 4.19iT - 37T^{2} \)
41 \( 1 - 11.7T + 41T^{2} \)
43 \( 1 + 2.64iT - 43T^{2} \)
47 \( 1 - 0.106iT - 47T^{2} \)
53 \( 1 - 7.86iT - 53T^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 - 10.0T + 61T^{2} \)
67 \( 1 + 1.28iT - 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + 11.8iT - 73T^{2} \)
79 \( 1 - 5.48T + 79T^{2} \)
83 \( 1 + 17.1iT - 83T^{2} \)
89 \( 1 + 4.31T + 89T^{2} \)
97 \( 1 - 7.65iT - 97T^{2} \)
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

−8.739978122277900263495364205178, −7.80610919129753028431737523730, −7.42814515909762260486832961625, −6.46423882060505179110477898541, −5.83070652431110125939533269536, −4.66906806953743767423274970979, −3.90255556638583005980735612961, −2.80538441568171406587557025781, −1.52676065782848001523055422943, −0.65380428371892207886212754969, 1.27379293387009218385449980060, 2.89142493213589924200608534267, 3.90917196727909052918837049855, 4.14906938161177984449929739177, 5.07112719619052551483292670207, 6.34299062648313688417475167674, 6.96529121965837237044836768027, 7.68764662395527225572426400530, 8.707476917613690516460861396242, 9.433545221854133790067991197975

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