↑

Properties

Label 2-2e10-32.13-c1-0-0
Degree $2$
Conductor $1024$
Sign $-0.195 + 0.980i$
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  + (−0.401 + 0.969i)3-s + (−3.49 + 1.44i)5-s + (−3.21 + 3.21i)7-s + (1.34 + 1.34i)9-s + (0.363 + 0.878i)11-s + (−2.37 − 0.985i)13-s − 3.97i·15-s − 1.34i·17-s + (−3.10 − 1.28i)19-s + (−1.82 − 4.41i)21-s + (4.30 + 4.30i)23-s + (6.60 − 6.60i)25-s + (−4.74 + 1.96i)27-s + (0.600 − 1.44i)29-s + 3.69·31-s + ⋯
L(s)  = 1  + (−0.231 + 0.559i)3-s + (−1.56 + 0.648i)5-s + (−1.21 + 1.21i)7-s + (0.447 + 0.447i)9-s + (0.109 + 0.264i)11-s + (−0.659 − 0.273i)13-s − 1.02i·15-s − 0.326i·17-s + (−0.712 − 0.295i)19-s + (−0.398 − 0.963i)21-s + (0.896 + 0.896i)23-s + (1.32 − 1.32i)25-s + (−0.914 + 0.378i)27-s + (0.111 − 0.269i)29-s + 0.663·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-0.195 + 0.980i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ -0.195 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1503556477\)
\(L(\frac12)\) \(\approx\) \(0.1503556477\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.401 - 0.969i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (3.49 - 1.44i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (3.21 - 3.21i)T - 7iT^{2} \)
11 \( 1 + (-0.363 - 0.878i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (2.37 + 0.985i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 1.34iT - 17T^{2} \)
19 \( 1 + (3.10 + 1.28i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-4.30 - 4.30i)T + 23iT^{2} \)
29 \( 1 + (-0.600 + 1.44i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 3.69T + 31T^{2} \)
37 \( 1 + (-4.03 + 1.67i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (5.34 + 5.34i)T + 41iT^{2} \)
43 \( 1 + (-3.01 - 7.27i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 1.02iT - 47T^{2} \)
53 \( 1 + (2.69 + 6.49i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (5.90 - 2.44i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-1.83 + 4.42i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (2.58 - 6.23i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (5.75 - 5.75i)T - 71iT^{2} \)
73 \( 1 + (2.94 + 2.94i)T + 73iT^{2} \)
79 \( 1 + 15.4iT - 79T^{2} \)
83 \( 1 + (-11.2 - 4.66i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (9.04 - 9.04i)T - 89iT^{2} \)
97 \( 1 - 8.41T + 97T^{2} \)
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

−10.53328233844430336228570991053, −9.738481849815787954177921890927, −9.017609995542256508996400349052, −7.956777885433307557146047802967, −7.20380592576328717894846954098, −6.42785607135459170484284015780, −5.24660906876457635682292127816, −4.35520558797955547043493035397, −3.36721119392008101949201595685, −2.56002120876339236988853369240, 0.086270936297765402751111999505, 1.04183585673116433680571898757, 3.14160073642511892410996956995, 4.06058164162395104436552807376, 4.59787356946490058775054700458, 6.26918024517563386265094832841, 6.92520971232851658592709824866, 7.51748425516866919497492972081, 8.384217197716599079405945915449, 9.314450249807335176461386773133

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