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Properties

Label 2-2e10-16.11-c0-0-0
Degree $2$
Conductor $1024$
Sign $0.382 - 0.923i$
Analytic cond. $0.511042$
Root an. cond. $0.714872$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)3-s + i·9-s + (−1 + i)11-s + (1 + i)19-s − i·25-s − 2·33-s − 2i·41-s + (1 − i)43-s − 49-s + 2i·57-s + (−1 + i)59-s + (−1 − i)67-s + (1 − i)75-s + 81-s + (−1 − i)83-s + ⋯
L(s)  = 1  + (1 + i)3-s + i·9-s + (−1 + i)11-s + (1 + i)19-s − i·25-s − 2·33-s − 2i·41-s + (1 − i)43-s − 49-s + 2i·57-s + (−1 + i)59-s + (−1 − i)67-s + (1 − i)75-s + 81-s + (−1 − i)83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(0.511042\)
Root analytic conductor: \(0.714872\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :0),\ 0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.347086661\)
\(L(\frac12)\) \(\approx\) \(1.347086661\)
\(L(1)\) 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 \( 1 + (-1 - i)T + iT^{2} \)
5 \( 1 + iT^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (1 - i)T - iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + 2iT - T^{2} \)
43 \( 1 + (-1 + i)T - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (1 - i)T - iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.23264703266386487940188096482, −9.520815582826900357019204434140, −8.770531152308037428147616145906, −7.896878669410550189139682898349, −7.26327532899938756307467989777, −5.85381950754256431730693884339, −4.88320686210421956169134443208, −4.06340453761782094194168322246, −3.10604574614144044981901000022, −2.11145094969115851311836062866, 1.30583557128706184342674545039, 2.74050997633500171799463602234, 3.20597135561559039584818918755, 4.79851693176477679705188100747, 5.81415762181919571934465477131, 6.84353244022132700337272567529, 7.68202773781370975790556099714, 8.144836820057251915859162225216, 9.027121900810979546490752749527, 9.748377319093632746949284888440

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