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Properties

Label 2-2e10-16.5-c1-0-24
Degree $2$
Conductor $1024$
Sign $-0.923 - 0.382i$
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  + (−1 + i)3-s + (−2 − 2i)5-s − 4i·7-s + i·9-s + (−1 − i)11-s + (2 − 2i)13-s + 4·15-s + 4·17-s + (−5 + 5i)19-s + (4 + 4i)21-s + 4i·23-s + 3i·25-s + (−4 − 4i)27-s + (−6 + 6i)29-s − 8·31-s + ⋯
L(s)  = 1  + (−0.577 + 0.577i)3-s + (−0.894 − 0.894i)5-s − 1.51i·7-s + 0.333i·9-s + (−0.301 − 0.301i)11-s + (0.554 − 0.554i)13-s + 1.03·15-s + 0.970·17-s + (−1.14 + 1.14i)19-s + (0.872 + 0.872i)21-s + 0.834i·23-s + 0.600i·25-s + (−0.769 − 0.769i)27-s + (−1.11 + 1.11i)29-s − 1.43·31-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: $-0.923 - 0.382i$
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: \((2,\ 1024,\ (\ :1/2),\ -0.923 - 0.382i)\)

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 \( 1 + (1 - i)T - 3iT^{2} \)
5 \( 1 + (2 + 2i)T + 5iT^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + (1 + i)T + 11iT^{2} \)
13 \( 1 + (-2 + 2i)T - 13iT^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + (5 - 5i)T - 19iT^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + (6 - 6i)T - 29iT^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (-2 - 2i)T + 37iT^{2} \)
41 \( 1 + 2iT - 41T^{2} \)
43 \( 1 + (3 + 3i)T + 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (2 + 2i)T + 53iT^{2} \)
59 \( 1 + (-3 - 3i)T + 59iT^{2} \)
61 \( 1 + (-6 + 6i)T - 61iT^{2} \)
67 \( 1 + (3 - 3i)T - 67iT^{2} \)
71 \( 1 - 4iT - 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (7 - 7i)T - 83iT^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−9.689360619551227449509741235991, −8.435780683320674384609438935994, −7.88737052994389186279896323492, −7.16217914091863556617504823248, −5.72812836812556639272719159386, −5.11623368886745793290367160927, −3.94921456251181557653383699816, −3.70290815934971434515936580770, −1.37179631026388951266522490264, 0, 2.02325812911198847053479850855, 3.09047139588442463285711751689, 4.19942307312353317363870580401, 5.52106267555180265475607362931, 6.23097237621964385416743819899, 6.96779062635909559457398390557, 7.77891682488063234786745697551, 8.755096790324712371861479237440, 9.447651134227729078341170471500

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