Properties

Label 2-2e10-1.1-c1-0-13
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.08·3-s + 0.585·5-s + 3.69·7-s − 1.82·9-s + 4.14·11-s + 3.41·13-s + 0.634·15-s + 2.82·17-s − 6.30·19-s + 4·21-s − 6.75·23-s − 4.65·25-s − 5.22·27-s + 7.41·29-s + 3.06·31-s + 4.48·33-s + 2.16·35-s + 9.07·37-s + 3.69·39-s − 4·41-s − 1.08·43-s − 1.07·45-s + 3.06·47-s + 6.65·49-s + 3.06·51-s + 4.58·53-s + 2.42·55-s + ⋯
L(s)  = 1  + 0.624·3-s + 0.261·5-s + 1.39·7-s − 0.609·9-s + 1.24·11-s + 0.946·13-s + 0.163·15-s + 0.685·17-s − 1.44·19-s + 0.872·21-s − 1.40·23-s − 0.931·25-s − 1.00·27-s + 1.37·29-s + 0.549·31-s + 0.780·33-s + 0.365·35-s + 1.49·37-s + 0.591·39-s − 0.624·41-s − 0.165·43-s − 0.159·45-s + 0.446·47-s + 0.950·49-s + 0.428·51-s + 0.629·53-s + 0.327·55-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

\(L(1)\) \(\approx\) \(2.455644790\)
\(L(\frac12)\) \(\approx\) \(2.455644790\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.08T + 3T^{2} \)
5 \( 1 - 0.585T + 5T^{2} \)
7 \( 1 - 3.69T + 7T^{2} \)
11 \( 1 - 4.14T + 11T^{2} \)
13 \( 1 - 3.41T + 13T^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 + 6.30T + 19T^{2} \)
23 \( 1 + 6.75T + 23T^{2} \)
29 \( 1 - 7.41T + 29T^{2} \)
31 \( 1 - 3.06T + 31T^{2} \)
37 \( 1 - 9.07T + 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 1.08T + 43T^{2} \)
47 \( 1 - 3.06T + 47T^{2} \)
53 \( 1 - 4.58T + 53T^{2} \)
59 \( 1 - 1.08T + 59T^{2} \)
61 \( 1 + 1.07T + 61T^{2} \)
67 \( 1 - 1.97T + 67T^{2} \)
71 \( 1 + 8.02T + 71T^{2} \)
73 \( 1 + 6.48T + 73T^{2} \)
79 \( 1 + 14.7T + 79T^{2} \)
83 \( 1 - 13.6T + 83T^{2} \)
89 \( 1 - 4.82T + 89T^{2} \)
97 \( 1 - 5.17T + 97T^{2} \)
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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

−9.901737894378749545138942968076, −8.853896894844756469920230056290, −8.355530115495223313565532289630, −7.79380577458599340032547642662, −6.38157768700889576918052761335, −5.82120843789372009944298462339, −4.47773709437678088260033584226, −3.78169473542701537862071289041, −2.37617883816608787264309684634, −1.39136315239127214852981826771, 1.39136315239127214852981826771, 2.37617883816608787264309684634, 3.78169473542701537862071289041, 4.47773709437678088260033584226, 5.82120843789372009944298462339, 6.38157768700889576918052761335, 7.79380577458599340032547642662, 8.355530115495223313565532289630, 8.853896894844756469920230056290, 9.901737894378749545138942968076

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