Properties

Label 2-2e10-32.29-c1-0-12
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: $\chi_{1024} (641, \cdot )$
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

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

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