↑

Properties

Label 2-1040-13.10-c1-0-6
Degree $2$
Conductor $1040$
Sign $0.943 - 0.331i$
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  + (−1.16 − 2.01i)3-s + i·5-s + (−0.346 − 0.199i)7-s + (−1.21 + 2.11i)9-s + (−1.5 + 0.866i)11-s + (−0.619 + 3.55i)13-s + (2.01 − 1.16i)15-s + (0.346 − 0.599i)17-s + (4.65 + 2.68i)19-s + 0.932i·21-s + (0.0535 + 0.0927i)23-s − 25-s − 1.30·27-s + (2.45 + 4.24i)29-s + 7.86i·31-s + ⋯
L(s)  = 1  + (−0.673 − 1.16i)3-s + 0.447i·5-s + (−0.130 − 0.0755i)7-s + (−0.406 + 0.704i)9-s + (−0.452 + 0.261i)11-s + (−0.171 + 0.985i)13-s + (0.521 − 0.301i)15-s + (0.0839 − 0.145i)17-s + (1.06 + 0.616i)19-s + 0.203i·21-s + (0.0111 + 0.0193i)23-s − 0.200·25-s − 0.251·27-s + (0.455 + 0.788i)29-s + 1.41i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.943 - 0.331i$
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: \((2,\ 1040,\ (\ :1/2),\ 0.943 - 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.033771381\)
\(L(\frac12)\) \(\approx\) \(1.033771381\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.619 - 3.55i)T \)
good3 \( 1 + (1.16 + 2.01i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.346 + 0.199i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.346 + 0.599i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.65 - 2.68i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.0535 - 0.0927i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.45 - 4.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.86iT - 31T^{2} \)
37 \( 1 + (-1.96 + 1.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.69 + 3.86i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.00 - 5.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 + (-6.30 - 3.63i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.34 - 7.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.15 + 0.664i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.35 - 1.93i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 - 14.0iT - 83T^{2} \)
89 \( 1 + (-0.300 + 0.173i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.66 + 4.42i)T + (48.5 + 84.0i)T^{2} \)
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

−10.07874766733242306420526596930, −9.163212255656725540390144687367, −8.053694941830470021103037594110, −7.15857554934625438118368334927, −6.83403292228619725945934773233, −5.84226847380944693264830402477, −4.99227620562832904718147767009, −3.61989240364203324163980617596, −2.33840140446612857214847514074, −1.17101090231423304932830792100, 0.59175805557544311847260399713, 2.66333642348239618020434134603, 3.81118550058604137667912455682, 4.76298028376355459173237297049, 5.44102195929899928829815452926, 6.11974749845173785059053449076, 7.52809814885091464234452826749, 8.246538810919991383892200803872, 9.419389259931681855590699355527, 9.821543286035339650873672389115

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