↑

Properties

Label 2-2e10-16.13-c1-0-18
Degree $2$
Conductor $1024$
Sign $0.382 + 0.923i$
Analytic cond. $8.17668$
Root an. cond. $2.85948$
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  + (0.414 + 0.414i)3-s − 2.65i·9-s + (4.41 − 4.41i)11-s − 5.65·17-s + (−5.24 − 5.24i)19-s + 5i·25-s + (2.34 − 2.34i)27-s + 3.65·33-s − 6i·41-s + (9.24 − 9.24i)43-s + 7·49-s + (−2.34 − 2.34i)51-s − 4.34i·57-s + (10.0 − 10.0i)59-s + (−2.75 − 2.75i)67-s + ⋯
L(s)  = 1  + (0.239 + 0.239i)3-s − 0.885i·9-s + (1.33 − 1.33i)11-s − 1.37·17-s + (−1.20 − 1.20i)19-s + i·25-s + (0.450 − 0.450i)27-s + 0.636·33-s − 0.937i·41-s + (1.40 − 1.40i)43-s + 49-s + (−0.328 − 0.328i)51-s − 0.575i·57-s + (1.31 − 1.31i)59-s + (−0.336 − 0.336i)67-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ 0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.569927204\)
\(L(\frac12)\) \(\approx\) \(1.569927204\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.414 - 0.414i)T + 3iT^{2} \)
5 \( 1 - 5iT^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + (-4.41 + 4.41i)T - 11iT^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + 5.65T + 17T^{2} \)
19 \( 1 + (5.24 + 5.24i)T + 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 + (-9.24 + 9.24i)T - 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + (-10.0 + 10.0i)T - 59iT^{2} \)
61 \( 1 + 61iT^{2} \)
67 \( 1 + (2.75 + 2.75i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 16.9iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-7.58 - 7.58i)T + 83iT^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + 16.9T + 97T^{2} \)
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

−9.513670689817273088821837112701, −8.851615081761290966180300062298, −8.605088742768966440454116072901, −6.99577023683221917324732323950, −6.53689413292957176462925808585, −5.57371255021932308273840259133, −4.20775849720437476177250311801, −3.65680691683731969713367555538, −2.37887918742734390799114219086, −0.70474602175357628673949078916, 1.66326741471204731763943248100, 2.47055208505877805003098898606, 4.18127571881760691028591190849, 4.53449822148307833126954668662, 6.03226073650220811979945246006, 6.74636434350181140817487464438, 7.59671215618770683628167631819, 8.465481872798565058968022205711, 9.195858241346748352463531695055, 10.12622387407484991725298404681

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