↑

Properties

Label 2-2e10-32.5-c1-0-16
Degree $2$
Conductor $1024$
Sign $0.980 - 0.195i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.942 + 2.27i)3-s + (−2.49 − 1.03i)5-s + (−1.37 − 1.37i)7-s + (−2.17 + 2.17i)9-s + (1.70 − 4.12i)11-s + (4.86 − 2.01i)13-s − 6.66i·15-s + 6.31i·17-s + (4.72 − 1.95i)19-s + (1.82 − 4.41i)21-s + (0.288 − 0.288i)23-s + (1.63 + 1.63i)25-s + (−0.159 − 0.0660i)27-s + (0.428 + 1.03i)29-s − 3.69·31-s + ⋯
L(s)  = 1  + (0.544 + 1.31i)3-s + (−1.11 − 0.462i)5-s + (−0.518 − 0.518i)7-s + (−0.723 + 0.723i)9-s + (0.515 − 1.24i)11-s + (1.34 − 0.558i)13-s − 1.72i·15-s + 1.53i·17-s + (1.08 − 0.449i)19-s + (0.398 − 0.963i)21-s + (0.0602 − 0.0602i)23-s + (0.327 + 0.327i)25-s + (−0.0306 − 0.0127i)27-s + (0.0796 + 0.192i)29-s − 0.663·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $0.980 - 0.195i$
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: \((2,\ 1024,\ (\ :1/2),\ 0.980 - 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.592020976\)
\(L(\frac12)\) \(\approx\) \(1.592020976\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.942 - 2.27i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (2.49 + 1.03i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (1.37 + 1.37i)T + 7iT^{2} \)
11 \( 1 + (-1.70 + 4.12i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-4.86 + 2.01i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 6.31iT - 17T^{2} \)
19 \( 1 + (-4.72 + 1.95i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-0.288 + 0.288i)T - 23iT^{2} \)
29 \( 1 + (-0.428 - 1.03i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 3.69T + 31T^{2} \)
37 \( 1 + (-10.4 - 4.32i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.68 + 1.68i)T - 41iT^{2} \)
43 \( 1 + (-1.67 + 4.03i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 4.83iT - 47T^{2} \)
53 \( 1 + (-0.207 + 0.501i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (7.25 + 3.00i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.40 + 3.39i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (0.123 + 0.299i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-6.54 - 6.54i)T + 71iT^{2} \)
73 \( 1 + (-5.53 + 5.53i)T - 73iT^{2} \)
79 \( 1 + 0.877iT - 79T^{2} \)
83 \( 1 + (-3.42 + 1.41i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-6.46 - 6.46i)T + 89iT^{2} \)
97 \( 1 + 3.58T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.929640444123692661940489879603, −9.043344583951703669476506308103, −8.446602538863382360765443382738, −7.86670149278277095857145690124, −6.49505877722952503178893085208, −5.54784080237797875557372109271, −4.31817266570394639069395681839, −3.61687354051567971991701238965, −3.33326928898893148182274231496, −0.865459075323616257462728658211, 1.17173877854830280520946771290, 2.48791210522614570283918441271, 3.41033333583343604196060017521, 4.45971021500813938195761887533, 5.98780368661630086158122196971, 6.80187752364150353556562243829, 7.49277443812576561912678900755, 7.890936234707903079633035094437, 9.131490935165693189389126521314, 9.537144524884837689870536213982

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