Properties

Label 2-2e10-1.1-c1-0-5
Degree $2$
Conductor $1024$
Sign $1$
Analytic cond. $8.17668$
Root an. cond. $2.85948$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.61·3-s + 3.41·5-s − 1.53·7-s + 3.82·9-s + 4.77·11-s + 0.585·13-s − 8.92·15-s − 2.82·17-s + 0.448·19-s + 4·21-s − 5.86·23-s + 6.65·25-s − 2.16·27-s + 4.58·29-s + 7.39·31-s − 12.4·33-s − 5.22·35-s − 5.07·37-s − 1.53·39-s − 4·41-s + 2.61·43-s + 13.0·45-s + 7.39·47-s − 4.65·49-s + 7.39·51-s + 7.41·53-s + 16.3·55-s + ⋯
L(s)  = 1  − 1.50·3-s + 1.52·5-s − 0.578·7-s + 1.27·9-s + 1.44·11-s + 0.162·13-s − 2.30·15-s − 0.685·17-s + 0.102·19-s + 0.872·21-s − 1.22·23-s + 1.33·25-s − 0.416·27-s + 0.851·29-s + 1.32·31-s − 2.17·33-s − 0.883·35-s − 0.833·37-s − 0.245·39-s − 0.624·41-s + 0.398·43-s + 1.94·45-s + 1.07·47-s − 0.665·49-s + 1.03·51-s + 1.01·53-s + 2.19·55-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.261448799\)
\(L(\frac12)\) \(\approx\) \(1.261448799\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.61T + 3T^{2} \)
5 \( 1 - 3.41T + 5T^{2} \)
7 \( 1 + 1.53T + 7T^{2} \)
11 \( 1 - 4.77T + 11T^{2} \)
13 \( 1 - 0.585T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 - 0.448T + 19T^{2} \)
23 \( 1 + 5.86T + 23T^{2} \)
29 \( 1 - 4.58T + 29T^{2} \)
31 \( 1 - 7.39T + 31T^{2} \)
37 \( 1 + 5.07T + 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 - 2.61T + 43T^{2} \)
47 \( 1 - 7.39T + 47T^{2} \)
53 \( 1 - 7.41T + 53T^{2} \)
59 \( 1 + 2.61T + 59T^{2} \)
61 \( 1 - 13.0T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 - 6.12T + 79T^{2} \)
83 \( 1 + 3.50T + 83T^{2} \)
89 \( 1 + 0.828T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.01151418964978744596009809977, −9.440896840295615116831456353651, −8.480430152060520178172646238244, −6.68372788424544699424761403689, −6.56165961618439836198526720966, −5.82626464540961838981061684282, −4.99352124326692809550077641306, −3.89480476744320597052986510397, −2.22630838807908370909208512995, −0.980969940577510576536911232947, 0.980969940577510576536911232947, 2.22630838807908370909208512995, 3.89480476744320597052986510397, 4.99352124326692809550077641306, 5.82626464540961838981061684282, 6.56165961618439836198526720966, 6.68372788424544699424761403689, 8.480430152060520178172646238244, 9.440896840295615116831456353651, 10.01151418964978744596009809977

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