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Properties

Label 2-2e10-16.13-c1-0-3
Degree $2$
Conductor $1024$
Sign $0.923 - 0.382i$
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  + (−1.41 − 1.41i)3-s + 1.00i·9-s + (−4.24 + 4.24i)11-s + 6·17-s + (1.41 + 1.41i)19-s + 5i·25-s + (−2.82 + 2.82i)27-s + 12·33-s + 6i·41-s + (7.07 − 7.07i)43-s + 7·49-s + (−8.48 − 8.48i)51-s − 4.00i·57-s + (4.24 − 4.24i)59-s + (9.89 + 9.89i)67-s + ⋯
L(s)  = 1  + (−0.816 − 0.816i)3-s + 0.333i·9-s + (−1.27 + 1.27i)11-s + 1.45·17-s + (0.324 + 0.324i)19-s + i·25-s + (−0.544 + 0.544i)27-s + 2.08·33-s + 0.937i·41-s + (1.07 − 1.07i)43-s + 49-s + (−1.18 − 1.18i)51-s − 0.529i·57-s + (0.552 − 0.552i)59-s + (1.20 + 1.20i)67-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $0.923 - 0.382i$
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: \((2,\ 1024,\ (\ :1/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9639516481\)
\(L(\frac12)\) \(\approx\) \(0.9639516481\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + 1.41i)T + 3iT^{2} \)
5 \( 1 - 5iT^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + (4.24 - 4.24i)T - 11iT^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + (-1.41 - 1.41i)T + 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 + (-7.07 + 7.07i)T - 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + (-4.24 + 4.24i)T - 59iT^{2} \)
61 \( 1 + 61iT^{2} \)
67 \( 1 + (-9.89 - 9.89i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-12.7 - 12.7i)T + 83iT^{2} \)
89 \( 1 - 18iT - 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−10.06743942912098032797778006029, −9.341416627930991817366332416053, −7.945807704405724987747617337304, −7.47720149499705179684040303567, −6.71868788365629328352265854086, −5.55996980629332679546250941179, −5.18443010156024110190770497192, −3.73536329035540728450402142563, −2.37779778058763256097626411826, −1.10358116217718477001623155277, 0.58446515049683610744232804831, 2.64159106041205597758567489954, 3.68802895396616860429037563767, 4.84573342570948769801696342456, 5.54906565480838406738227109189, 6.11175974453244368623358438975, 7.53930086823353666922259782383, 8.172346446678142393497450473084, 9.206342830987436660234100973306, 10.24029579333270747511705414468

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