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Properties

Label 2-2e10-32.21-c1-0-3
Degree $2$
Conductor $1024$
Sign $-0.659 - 0.751i$
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  + (0.860 − 0.356i)3-s + (−0.366 + 0.883i)5-s + (−2.35 − 2.35i)7-s + (−1.50 + 1.50i)9-s + (−1.81 − 0.752i)11-s + (−0.848 − 2.04i)13-s + 0.891i·15-s + 6.00i·17-s + (1.54 + 3.73i)19-s + (−2.86 − 1.18i)21-s + (−2.91 + 2.91i)23-s + (2.88 + 2.88i)25-s + (−1.82 + 4.41i)27-s + (−9.69 + 4.01i)29-s + 7.52·31-s + ⋯
L(s)  = 1  + (0.497 − 0.205i)3-s + (−0.163 + 0.395i)5-s + (−0.889 − 0.889i)7-s + (−0.502 + 0.502i)9-s + (−0.547 − 0.226i)11-s + (−0.235 − 0.568i)13-s + 0.230i·15-s + 1.45i·17-s + (0.354 + 0.856i)19-s + (−0.624 − 0.258i)21-s + (−0.607 + 0.607i)23-s + (0.577 + 0.577i)25-s + (−0.352 + 0.850i)27-s + (−1.80 + 0.745i)29-s + 1.35·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-0.659 - 0.751i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ -0.659 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5747376711\)
\(L(\frac12)\) \(\approx\) \(0.5747376711\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.860 + 0.356i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.366 - 0.883i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (2.35 + 2.35i)T + 7iT^{2} \)
11 \( 1 + (1.81 + 0.752i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.848 + 2.04i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 6.00iT - 17T^{2} \)
19 \( 1 + (-1.54 - 3.73i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (2.91 - 2.91i)T - 23iT^{2} \)
29 \( 1 + (9.69 - 4.01i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 7.52T + 31T^{2} \)
37 \( 1 + (0.994 - 2.40i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.37 - 1.37i)T - 41iT^{2} \)
43 \( 1 + (10.8 + 4.50i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 3.33iT - 47T^{2} \)
53 \( 1 + (8.02 + 3.32i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-4.70 + 11.3i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (0.166 - 0.0688i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-8.39 + 3.47i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (7.92 + 7.92i)T + 71iT^{2} \)
73 \( 1 + (5.84 - 5.84i)T - 73iT^{2} \)
79 \( 1 + 1.80iT - 79T^{2} \)
83 \( 1 + (1.98 + 4.79i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-6.38 - 6.38i)T + 89iT^{2} \)
97 \( 1 - 0.874T + 97T^{2} \)
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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

−10.30712358721898888583616786551, −9.545718058753005133056049668673, −8.309290562634264811019239427800, −7.87231140245504356912966500701, −7.01010526925688518759374667073, −6.06035518945113675223798075718, −5.13628519129479770604971505796, −3.57987835403148059460509889026, −3.25210244093844011602864680160, −1.77284393665414699230227106879, 0.22726660910837873341283456252, 2.44545344294708756268026450262, 3.03330054126576609779149521196, 4.32688433448273278042775610440, 5.27312891541822533075434354681, 6.23977582489788952209979634281, 7.10521867014886861327550744119, 8.184414290441472839736547957350, 8.987422182601115813139087476094, 9.455331821156869906469757633352

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