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Properties

Label 2-2e10-4.3-c0-0-1
Degree $2$
Conductor $1024$
Sign $1$
Analytic cond. $0.511042$
Root an. cond. $0.714872$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·5-s + 9-s − 1.41·13-s + 1.00·25-s − 1.41·29-s − 1.41·37-s + 1.41·45-s + 49-s + 1.41·53-s + 1.41·61-s − 2.00·65-s − 2·73-s + 81-s − 2·89-s − 1.41·101-s + 1.41·109-s − 2·113-s − 1.41·117-s + ⋯
L(s)  = 1  + 1.41·5-s + 9-s − 1.41·13-s + 1.00·25-s − 1.41·29-s − 1.41·37-s + 1.41·45-s + 49-s + 1.41·53-s + 1.41·61-s − 2.00·65-s − 2·73-s + 81-s − 2·89-s − 1.41·101-s + 1.41·109-s − 2·113-s − 1.41·117-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $1$
Analytic conductor: \(0.511042\)
Root analytic conductor: \(0.714872\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (1023, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.287212050\)
\(L(\frac12)\) \(\approx\) \(1.287212050\)
\(L(1)\) 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 \( 1 - T^{2} \)
5 \( 1 - 1.41T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 1.41T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.41T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 2T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 2T + T^{2} \)
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

−9.997899310617523412848206287414, −9.561505476420194417584751785779, −8.692705219884471130321709336633, −7.36930641818480273493423131899, −6.91746467015679374490269786164, −5.73872637613625917760317193959, −5.11870657883250598875592977004, −4.00532088203678843236479581500, −2.53007454208825221301291880587, −1.66563761907292491067644962538, 1.66563761907292491067644962538, 2.53007454208825221301291880587, 4.00532088203678843236479581500, 5.11870657883250598875592977004, 5.73872637613625917760317193959, 6.91746467015679374490269786164, 7.36930641818480273493423131899, 8.692705219884471130321709336633, 9.561505476420194417584751785779, 9.997899310617523412848206287414

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