Properties

Label 2-2e10-1.1-c1-0-18
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  + 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

\(L(1)\) \(\approx\) \(3.349497179\)
\(L(\frac12)\) \(\approx\) \(3.349497179\)
\(L(\frac{3}{2})\) 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 \( 1 - 10.8T + 97T^{2} \)
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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

−9.911209800559379005419190548546, −8.899236700867210755173476875946, −8.579113887144332462271127008621, −7.59235694944000313718768258554, −6.71760415623890164988738747847, −5.48868131257463684832976022267, −4.80382925134991968340577962083, −3.30048009715135976557459575386, −2.42991404686427314299510988882, −1.73155908422959962028865206419, 1.73155908422959962028865206419, 2.42991404686427314299510988882, 3.30048009715135976557459575386, 4.80382925134991968340577962083, 5.48868131257463684832976022267, 6.71760415623890164988738747847, 7.59235694944000313718768258554, 8.579113887144332462271127008621, 8.899236700867210755173476875946, 9.911209800559379005419190548546

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