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Properties

Label 2-2e10-16.5-c1-0-19
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 2i)3-s − 5i·9-s + (2 + 2i)11-s − 6·17-s + (−6 + 6i)19-s − 5i·25-s + (4 + 4i)27-s − 8·33-s − 6i·41-s + (−6 − 6i)43-s + 7·49-s + (12 − 12i)51-s − 24i·57-s + (−10 − 10i)59-s + (6 − 6i)67-s + ⋯
L(s)  = 1  + (−1.15 + 1.15i)3-s − 1.66i·9-s + (0.603 + 0.603i)11-s − 1.45·17-s + (−1.37 + 1.37i)19-s − i·25-s + (0.769 + 0.769i)27-s − 1.39·33-s − 0.937i·41-s + (−0.914 − 0.914i)43-s + 49-s + (1.68 − 1.68i)51-s − 3.17i·57-s + (−1.30 − 1.30i)59-s + (0.733 − 0.733i)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: $\chi_{1024} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ -0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 2i)T - 3iT^{2} \)
5 \( 1 + 5iT^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + (-2 - 2i)T + 11iT^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + (6 - 6i)T - 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 + (6 + 6i)T + 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + (10 + 10i)T + 59iT^{2} \)
61 \( 1 - 61iT^{2} \)
67 \( 1 + (-6 + 6i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-2 + 2i)T - 83iT^{2} \)
89 \( 1 - 18iT - 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.864817774256780506140088140240, −9.079333142672526382928142319300, −8.200735361291123137510445775522, −6.75172124706229530752788295853, −6.26531678317718414873911810734, −5.25944459715080340930668615307, −4.33378027211651351784423619181, −3.87492803420020889319438857115, −2.04790788539260179132268842795, 0, 1.36549827710483330589193810009, 2.58963681729570994937645386150, 4.23672589684993170572700363073, 5.17226344302626082840278043131, 6.27229838899994366656850162432, 6.61326704609483034513883321525, 7.42650399309919736148654323524, 8.537887679111114423380317944395, 9.203443160706563821878988179406

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