Properties

Label 2-2240-40.29-c1-0-18
Degree $2$
Conductor $2240$
Sign $0.403 - 0.915i$
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  − 3.11·3-s + (2.08 − 0.809i)5-s i·7-s + 6.68·9-s + 2.64i·11-s − 0.479·13-s + (−6.48 + 2.52i)15-s − 1.49i·17-s + 2.16i·19-s + 3.11i·21-s + 6.33i·23-s + (3.68 − 3.37i)25-s − 11.4·27-s + 5.25i·29-s + 6.22·31-s + ⋯
L(s)  = 1  − 1.79·3-s + (0.932 − 0.362i)5-s − 0.377i·7-s + 2.22·9-s + 0.798i·11-s − 0.133·13-s + (−1.67 + 0.650i)15-s − 0.362i·17-s + 0.497i·19-s + 0.679i·21-s + 1.32i·23-s + (0.737 − 0.675i)25-s − 2.20·27-s + 0.976i·29-s + 1.11·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.403 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.403 - 0.915i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.403 - 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8695065434\)
\(L(\frac12)\) \(\approx\) \(0.8695065434\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.08 + 0.809i)T \)
7 \( 1 + iT \)
good3 \( 1 + 3.11T + 3T^{2} \)
11 \( 1 - 2.64iT - 11T^{2} \)
13 \( 1 + 0.479T + 13T^{2} \)
17 \( 1 + 1.49iT - 17T^{2} \)
19 \( 1 - 2.16iT - 19T^{2} \)
23 \( 1 - 6.33iT - 23T^{2} \)
29 \( 1 - 5.25iT - 29T^{2} \)
31 \( 1 - 6.22T + 31T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 9.98T + 43T^{2} \)
47 \( 1 + 7.68iT - 47T^{2} \)
53 \( 1 - 3.04T + 53T^{2} \)
59 \( 1 + 9.20iT - 59T^{2} \)
61 \( 1 - 10.8iT - 61T^{2} \)
67 \( 1 - 9.73T + 67T^{2} \)
71 \( 1 + 0.525T + 71T^{2} \)
73 \( 1 - 9.98iT - 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 - 2.95T + 89T^{2} \)
97 \( 1 - 7.97iT - 97T^{2} \)
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

−9.538233361567427397232408475514, −8.433870301006844561969932489031, −7.16287069167155681596002124492, −6.83498260826016790002408867916, −5.92259874137407159101732230425, −5.17104510957454126177397750065, −4.84310673338292882781689426421, −3.65655575339994851223806194054, −1.93198643074126320584144736762, −1.06794281046309243682185568242, 0.44989977356344399249084398918, 1.73292258183391366834550204591, 2.95363982156678542126920264343, 4.36342108631894984517149048659, 5.15910910029440015364352196750, 5.78952053951487380953800692135, 6.47183021672417741046118318243, 6.81730285934520629756900290176, 8.105157031827030715877151507153, 9.028686646547542610333871204514

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