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Properties

Label 2-2e10-16.13-c1-0-21
Degree $2$
Conductor $1024$
Sign $-0.382 + 0.923i$
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 + i)3-s + (−1.41 + 1.41i)5-s − 2.82i·7-s − i·9-s + (−3 + 3i)11-s + (−4.24 − 4.24i)13-s − 2.82·15-s + (−3 − 3i)19-s + (2.82 − 2.82i)21-s − 8.48i·23-s + 0.999i·25-s + (4 − 4i)27-s + (1.41 + 1.41i)29-s − 5.65·31-s − 6·33-s + ⋯
L(s)  = 1  + (0.577 + 0.577i)3-s + (−0.632 + 0.632i)5-s − 1.06i·7-s − 0.333i·9-s + (−0.904 + 0.904i)11-s + (−1.17 − 1.17i)13-s − 0.730·15-s + (−0.688 − 0.688i)19-s + (0.617 − 0.617i)21-s − 1.76i·23-s + 0.199i·25-s + (0.769 − 0.769i)27-s + (0.262 + 0.262i)29-s − 1.01·31-s − 1.04·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ -0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6497106607\)
\(L(\frac12)\) \(\approx\) \(0.6497106607\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - i)T + 3iT^{2} \)
5 \( 1 + (1.41 - 1.41i)T - 5iT^{2} \)
7 \( 1 + 2.82iT - 7T^{2} \)
11 \( 1 + (3 - 3i)T - 11iT^{2} \)
13 \( 1 + (4.24 + 4.24i)T + 13iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (3 + 3i)T + 19iT^{2} \)
23 \( 1 + 8.48iT - 23T^{2} \)
29 \( 1 + (-1.41 - 1.41i)T + 29iT^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + (4.24 - 4.24i)T - 37iT^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 + (-3 + 3i)T - 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (1.41 - 1.41i)T - 53iT^{2} \)
59 \( 1 + (-1 + i)T - 59iT^{2} \)
61 \( 1 + (-4.24 - 4.24i)T + 61iT^{2} \)
67 \( 1 + (9 + 9i)T + 67iT^{2} \)
71 \( 1 + 8.48iT - 71T^{2} \)
73 \( 1 + 12iT - 73T^{2} \)
79 \( 1 + 5.65T + 79T^{2} \)
83 \( 1 + (-3 - 3i)T + 83iT^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
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

−9.919926715332829093980415873557, −8.873618515173004004028001317620, −7.88033403367057572097885337220, −7.32111631200078988370556800629, −6.56759630253006999708736229287, −4.98224043882081499273588639211, −4.34510463899704970871075080693, −3.29528251278729149908769956463, −2.52911573737214501175464027477, −0.25396265857524782218287560554, 1.84028367548830190118820019187, 2.66434844265036883738917235574, 3.94853392050716652169636129980, 5.12251396295358948507320556285, 5.76966196402896208117643116860, 7.15963674262036037375352584900, 7.77442958490949386536258235518, 8.579205343574030767160707663554, 9.021921117602624111412991319558, 10.08961106138684974268592874927

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