↑

Properties

Label 2-1040-13.10-c1-0-5
Degree $2$
Conductor $1040$
Sign $-0.265 - 0.964i$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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  + (0.366 + 0.633i)3-s + i·5-s + (−2.59 − 1.5i)7-s + (1.23 − 2.13i)9-s + (−2.59 + 1.5i)11-s + (3.5 + 0.866i)13-s + (−0.633 + 0.366i)15-s + (−4.09 + 7.09i)17-s + (0.401 + 0.232i)19-s − 2.19i·21-s + (4.73 + 8.19i)23-s − 25-s + 4·27-s + (1.26 + 2.19i)29-s + 4.73i·31-s + ⋯
L(s)  = 1  + (0.211 + 0.366i)3-s + 0.447i·5-s + (−0.981 − 0.566i)7-s + (0.410 − 0.711i)9-s + (−0.783 + 0.452i)11-s + (0.970 + 0.240i)13-s + (−0.163 + 0.0945i)15-s + (−0.993 + 1.72i)17-s + (0.0922 + 0.0532i)19-s − 0.479i·21-s + (0.986 + 1.70i)23-s − 0.200·25-s + 0.769·27-s + (0.235 + 0.407i)29-s + 0.849i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.265 - 0.964i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ -0.265 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.197108040\)
\(L(\frac12)\) \(\approx\) \(1.197108040\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (-3.5 - 0.866i)T \)
good3 \( 1 + (-0.366 - 0.633i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.59 + 1.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (4.09 - 7.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.401 - 0.232i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.73 - 8.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.26 - 2.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.73iT - 31T^{2} \)
37 \( 1 + (0.696 - 0.401i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (9 - 5.19i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 - 0.464T + 53T^{2} \)
59 \( 1 + (9 + 5.19i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.09 - 5.36i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.19 - 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 11.6iT - 73T^{2} \)
79 \( 1 - 4.19T + 79T^{2} \)
83 \( 1 + 8.19iT - 83T^{2} \)
89 \( 1 + (-5.89 + 3.40i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.90 - 4.56i)T + (48.5 + 84.0i)T^{2} \)
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

−10.31482333548395350148325467954, −9.370649503076165544336546041928, −8.700745697440532716350945294696, −7.57690392382622343725046997594, −6.68691676181667507311216926048, −6.19179922846839606642764548718, −4.82128859564282290914470308450, −3.67216534060348878440151435883, −3.27289180049381840529714993327, −1.55902382002382094108792899104, 0.52893395455188161064031907889, 2.32364198573927201060131854062, 3.06321500497633748730980651939, 4.52961724630950681681669507686, 5.33549530626607190735169286313, 6.38453191167161636912746174753, 7.11041791473494468128114133687, 8.176328881032347682370367405185, 8.788176166596856604078675960224, 9.546957780173051271452314464056

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