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Properties

Label 2-2e10-16.5-c1-0-25
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  + (2.41 − 2.41i)3-s − 8.65i·9-s + (−1.58 − 1.58i)11-s + 5.65·17-s + (−3.24 + 3.24i)19-s − 5i·25-s + (−13.6 − 13.6i)27-s − 7.65·33-s + 6i·41-s + (−0.757 − 0.757i)43-s + 7·49-s + (13.6 − 13.6i)51-s + 15.6i·57-s + (4.07 + 4.07i)59-s + (11.2 − 11.2i)67-s + ⋯
L(s)  = 1  + (1.39 − 1.39i)3-s − 2.88i·9-s + (−0.478 − 0.478i)11-s + 1.37·17-s + (−0.743 + 0.743i)19-s − i·25-s + (−2.62 − 2.62i)27-s − 1.33·33-s + 0.937i·41-s + (−0.115 − 0.115i)43-s + 49-s + (1.91 − 1.91i)51-s + 2.07i·57-s + (0.530 + 0.530i)59-s + (1.37 − 1.37i)67-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: $\chi_{1024} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ -0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.439312032\)
\(L(\frac12)\) \(\approx\) \(2.439312032\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.41 + 2.41i)T - 3iT^{2} \)
5 \( 1 + 5iT^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + (1.58 + 1.58i)T + 11iT^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 - 5.65T + 17T^{2} \)
19 \( 1 + (3.24 - 3.24i)T - 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 + (0.757 + 0.757i)T + 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + (-4.07 - 4.07i)T + 59iT^{2} \)
61 \( 1 - 61iT^{2} \)
67 \( 1 + (-11.2 + 11.2i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 16.9iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (10.4 - 10.4i)T - 83iT^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−9.537525339060746638410545623567, −8.427554475272526543570415295423, −8.182592782516335338772803243343, −7.36325230891039563569242765971, −6.48762645448488242350152721201, −5.67159693748965447920036818248, −3.97796192905610402633287236718, −3.07738617738081200031283938845, −2.18748628222973806491257955030, −0.971210703339165640438666208649, 2.07806965636099952056928833745, 3.06512548395131994316475455167, 3.87672290462991070377525885480, 4.81332458188226999915422146525, 5.54700496435849103646447532865, 7.22014922203102657513434712027, 7.890455861582382779674214193269, 8.737978709002334505687983434901, 9.341041811889180854583046139800, 10.15043835250896018521857941168

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