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Properties

Label 2-2e10-32.21-c1-0-14
Degree $2$
Conductor $1024$
Sign $0.442 - 0.896i$
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  + (−2.70 + 1.12i)3-s + (−0.151 + 0.366i)5-s + (3.06 + 3.06i)7-s + (3.95 − 3.95i)9-s + (3.66 + 1.51i)11-s + (−0.780 − 1.88i)13-s − 1.16i·15-s − 4.54i·17-s + (−0.221 − 0.534i)19-s + (−11.7 − 4.86i)21-s + (4.41 − 4.41i)23-s + (3.42 + 3.42i)25-s + (−2.91 + 7.03i)27-s + (4.74 − 1.96i)29-s + 0.0539·31-s + ⋯
L(s)  = 1  + (−1.56 + 0.647i)3-s + (−0.0678 + 0.163i)5-s + (1.15 + 1.15i)7-s + (1.31 − 1.31i)9-s + (1.10 + 0.457i)11-s + (−0.216 − 0.522i)13-s − 0.299i·15-s − 1.10i·17-s + (−0.0508 − 0.122i)19-s + (−2.56 − 1.06i)21-s + (0.921 − 0.921i)23-s + (0.684 + 0.684i)25-s + (−0.560 + 1.35i)27-s + (0.881 − 0.365i)29-s + 0.00969·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $0.442 - 0.896i$
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: \((2,\ 1024,\ (\ :1/2),\ 0.442 - 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.158867280\)
\(L(\frac12)\) \(\approx\) \(1.158867280\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.70 - 1.12i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.151 - 0.366i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-3.06 - 3.06i)T + 7iT^{2} \)
11 \( 1 + (-3.66 - 1.51i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.780 + 1.88i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 4.54iT - 17T^{2} \)
19 \( 1 + (0.221 + 0.534i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-4.41 + 4.41i)T - 23iT^{2} \)
29 \( 1 + (-4.74 + 1.96i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 0.0539T + 31T^{2} \)
37 \( 1 + (-0.330 + 0.798i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.621 + 0.621i)T - 41iT^{2} \)
43 \( 1 + (2.06 + 0.857i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 9.44iT - 47T^{2} \)
53 \( 1 + (-10.0 - 4.16i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (2.97 - 7.17i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-9.72 + 4.02i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (7.53 - 3.12i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (2.99 + 2.99i)T + 71iT^{2} \)
73 \( 1 + (2.91 - 2.91i)T - 73iT^{2} \)
79 \( 1 + 5.74iT - 79T^{2} \)
83 \( 1 + (-1.36 - 3.30i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-2.38 - 2.38i)T + 89iT^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−10.28443428244983076408910628846, −9.314586269808644412529818897315, −8.681482108284865224576575308858, −7.36204991166992527869378264584, −6.53384436056470791120141830973, −5.62072018880143126338034368898, −4.93272932775324787927052892083, −4.37125013529237407984853851331, −2.71769956209730124581231238909, −1.07584810737702601775278411457, 0.915678475219895319754870054895, 1.61667918743200463466502080484, 3.86717899732017880094937604749, 4.68166169700868026023608937583, 5.47964510224808750039244007045, 6.59575053168018883508123549187, 6.96307532808063482072678098298, 7.969027933945117041650993321407, 8.832298298579239164999735189217, 10.23768237158434173461423017363

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