Properties

Degree $2$
Conductor $1024$
Sign $0.923 - 0.382i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 1.41i)3-s + (1.41 + 1.41i)5-s + 4i·7-s − 1.00i·9-s + (−1.41 − 1.41i)11-s + (1.41 − 1.41i)13-s + 4.00·15-s + 2·17-s + (−1.41 + 1.41i)19-s + (5.65 + 5.65i)21-s + 4i·23-s − 0.999i·25-s + (2.82 + 2.82i)27-s + (4.24 − 4.24i)29-s − 4.00·33-s + ⋯
L(s)  = 1  + (0.816 − 0.816i)3-s + (0.632 + 0.632i)5-s + 1.51i·7-s − 0.333i·9-s + (−0.426 − 0.426i)11-s + (0.392 − 0.392i)13-s + 1.03·15-s + 0.485·17-s + (−0.324 + 0.324i)19-s + (1.23 + 1.23i)21-s + 0.834i·23-s − 0.199i·25-s + (0.544 + 0.544i)27-s + (0.787 − 0.787i)29-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $0.923 - 0.382i$
Motivic weight: \(1\)
Character: $\chi_{1024} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.345013099\)
\(L(\frac12)\) \(\approx\) \(2.345013099\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-1.41 + 1.41i)T - 3iT^{2} \)
5 \( 1 + (-1.41 - 1.41i)T + 5iT^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + (1.41 + 1.41i)T + 11iT^{2} \)
13 \( 1 + (-1.41 + 1.41i)T - 13iT^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + (1.41 - 1.41i)T - 19iT^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + (-4.24 + 4.24i)T - 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-7.07 - 7.07i)T + 37iT^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 + (-4.24 - 4.24i)T + 43iT^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + (-4.24 - 4.24i)T + 53iT^{2} \)
59 \( 1 + (9.89 + 9.89i)T + 59iT^{2} \)
61 \( 1 + (1.41 - 1.41i)T - 61iT^{2} \)
67 \( 1 + (-7.07 + 7.07i)T - 67iT^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + 14iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (-4.24 + 4.24i)T - 83iT^{2} \)
89 \( 1 + 2iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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

−9.831823326973790446938575509174, −9.098806193804102531797053295252, −8.056407383876123896830699747140, −7.919510699779543160589495179527, −6.37714583895726910445856578997, −6.04929189231026863666023906150, −4.94674293624403602174610833580, −3.11869155992713791672063183052, −2.65285222605664471768876986082, −1.63751757047662800139132892455, 1.06757726028612270039533786595, 2.56983300073975949157135885651, 3.83788875370598494527992325940, 4.37316585694665343858710972664, 5.35633980321687366441404325791, 6.62333895088324518299342854291, 7.45948193348215694965451488623, 8.474509505206986133625904796930, 9.105151362839531963921654237869, 9.936203449858407066101766210083

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