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Properties

Label 2-2e10-8.5-c1-0-2
Degree $2$
Conductor $1024$
Sign $-i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·5-s + 3·9-s + 7.07i·13-s − 8·17-s + 2.99·25-s − 4.24i·29-s + 9.89i·37-s + 8·41-s + 4.24i·45-s − 7·49-s + 12.7i·53-s + 15.5i·61-s − 10.0·65-s − 6·73-s + 9·81-s + ⋯
L(s)  = 1  + 0.632i·5-s + 9-s + 1.96i·13-s − 1.94·17-s + 0.599·25-s − 0.787i·29-s + 1.62i·37-s + 1.24·41-s + 0.632i·45-s − 49-s + 1.74i·53-s + 1.99i·61-s − 1.24·65-s − 0.702·73-s + 81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (513, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.429687019\)
\(L(\frac12)\) \(\approx\) \(1.429687019\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T^{2} \)
5 \( 1 - 1.41iT - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 7.07iT - 13T^{2} \)
17 \( 1 + 8T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 4.24iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 9.89iT - 37T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 12.7iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 15.5iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−10.13217898193172963257822181326, −9.314310229915961414502384416886, −8.672764413395070798897038849820, −7.39682947434747951407306110318, −6.77561244584082396014582566263, −6.22609404552281807143423112678, −4.50874480093786555185495387665, −4.26740037117766695282554928634, −2.69889583061787347538234318137, −1.65334165339279152226966748706, 0.66015831564846254449894922385, 2.12820322088497769881084109262, 3.46492958112130918888057512261, 4.57312390188671317000784340696, 5.25359728058462776446139238738, 6.37080064595568255930670415626, 7.27373285977552829048019520380, 8.119433769299278459419774122301, 8.911885291963261747923394328740, 9.688603366656961642275273958425

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