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Properties

Label 2-2e10-32.5-c1-0-1
Degree $2$
Conductor $1024$
Sign $-0.555 - 0.831i$
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  + (−0.660 − 1.59i)3-s + (−0.512 − 0.212i)5-s + (1.69 + 1.69i)7-s + (0.0177 − 0.0177i)9-s + (−1.42 + 3.44i)11-s + (−5.65 + 2.34i)13-s + 0.956i·15-s − 5.26i·17-s + (−3.80 + 1.57i)19-s + (1.58 − 3.82i)21-s + (−4.31 + 4.31i)23-s + (−3.31 − 3.31i)25-s + (−4.82 − 1.99i)27-s + (−0.512 − 1.23i)29-s + 1.53·31-s + ⋯
L(s)  = 1  + (−0.381 − 0.920i)3-s + (−0.229 − 0.0949i)5-s + (0.642 + 0.642i)7-s + (0.00590 − 0.00590i)9-s + (−0.429 + 1.03i)11-s + (−1.56 + 0.649i)13-s + 0.247i·15-s − 1.27i·17-s + (−0.871 + 0.361i)19-s + (0.346 − 0.835i)21-s + (−0.899 + 0.899i)23-s + (−0.663 − 0.663i)25-s + (−0.927 − 0.384i)27-s + (−0.0951 − 0.229i)29-s + 0.274·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-0.555 - 0.831i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ -0.555 - 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2632034626\)
\(L(\frac12)\) \(\approx\) \(0.2632034626\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.660 + 1.59i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (0.512 + 0.212i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.69 - 1.69i)T + 7iT^{2} \)
11 \( 1 + (1.42 - 3.44i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (5.65 - 2.34i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 5.26iT - 17T^{2} \)
19 \( 1 + (3.80 - 1.57i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (4.31 - 4.31i)T - 23iT^{2} \)
29 \( 1 + (0.512 + 1.23i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 1.53T + 31T^{2} \)
37 \( 1 + (-3.61 - 1.49i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (8.69 - 8.69i)T - 41iT^{2} \)
43 \( 1 + (-0.211 + 0.511i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 9.73iT - 47T^{2} \)
53 \( 1 + (-2.78 + 6.73i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (2.76 + 1.14i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-3.06 - 7.40i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-2.20 - 5.33i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (9.45 + 9.45i)T + 71iT^{2} \)
73 \( 1 + (1.69 - 1.69i)T - 73iT^{2} \)
79 \( 1 - 12.8iT - 79T^{2} \)
83 \( 1 + (-0.870 + 0.360i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (1.14 + 1.14i)T + 89iT^{2} \)
97 \( 1 + 1.83T + 97T^{2} \)
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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

−9.895044627009672532609663691913, −9.682018154471815041568107062261, −8.323808471419099842180405529415, −7.58495738757223103015450836233, −7.03595734894772332181149121131, −6.06094961561110582862263287215, −4.99767500516885967487314189024, −4.32732906723493152307466758359, −2.49558851151976506401817533053, −1.76383673368148434643437213874, 0.11651496757970940499333489553, 2.13080986228803551091875349390, 3.58515709964329435552922860560, 4.39677470395805195393118744929, 5.17335995111185288460861192534, 6.03625575429995200519100254545, 7.30850517053684543604259575165, 7.997562039576094594557173243661, 8.790302484822327377577393665297, 10.07389142884829233813286836868

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