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Properties

Label 2-2e10-16.11-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(0.6778226499\)
\(L(\frac12)\) \(\approx\) \(0.6778226499\)
\(L(1)\) 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

−9.999347270947810650710732954635, −8.888470298947247881066372007976, −8.239963597015327632430412186122, −7.04605459443902816807644445544, −6.51123595380878484907867711481, −5.86887145092419602923905958413, −4.81541429908606286467278580834, −3.61938485486740976888342394665, −2.12724089665506036130424914440, −0.74651767266997155423566748177, 1.76184800414805824537129392901, 3.56336289439858577351638330736, 4.37287304260647453232789367444, 5.10486705378116646183081955915, 6.10311149392121676345797317010, 6.78969302097901950716509682588, 7.931355097345279118608639886714, 9.015001255136887803843165878584, 9.819753436577072719303770472481, 10.28062805154763077499277304587

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