↑

Properties

Label 2-2e10-32.29-c1-0-21
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  + (2.90 + 1.20i)3-s + (−1.21 − 2.92i)5-s + (0.933 − 0.933i)7-s + (4.84 + 4.84i)9-s + (1.05 − 0.435i)11-s + (−1.09 + 2.65i)13-s − 9.94i·15-s − 1.61i·17-s + (2.34 − 5.66i)19-s + (3.82 − 1.58i)21-s + (1.67 + 1.67i)23-s + (−3.56 + 3.56i)25-s + (4.62 + 11.1i)27-s + (7.06 + 2.92i)29-s + 1.53·31-s + ⋯
L(s)  = 1  + (1.67 + 0.693i)3-s + (−0.542 − 1.30i)5-s + (0.352 − 0.352i)7-s + (1.61 + 1.61i)9-s + (0.317 − 0.131i)11-s + (−0.304 + 0.735i)13-s − 2.56i·15-s − 0.390i·17-s + (0.538 − 1.29i)19-s + (0.835 − 0.346i)21-s + (0.350 + 0.350i)23-s + (−0.712 + 0.712i)25-s + (0.890 + 2.15i)27-s + (1.31 + 0.543i)29-s + 0.274·31-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: $\chi_{1024} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ 0.980 + 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.790233689\)
\(L(\frac12)\) \(\approx\) \(2.790233689\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.90 - 1.20i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (1.21 + 2.92i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-0.933 + 0.933i)T - 7iT^{2} \)
11 \( 1 + (-1.05 + 0.435i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.09 - 2.65i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 1.61iT - 17T^{2} \)
19 \( 1 + (-2.34 + 5.66i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.67 - 1.67i)T + 23iT^{2} \)
29 \( 1 + (-7.06 - 2.92i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 1.53T + 31T^{2} \)
37 \( 1 + (2.08 + 5.04i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (1.03 + 1.03i)T + 41iT^{2} \)
43 \( 1 + (3.98 - 1.64i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 2.97iT - 47T^{2} \)
53 \( 1 + (-10.1 + 4.21i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (3.34 + 8.08i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (12.6 + 5.23i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (11.8 + 4.91i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (5.88 - 5.88i)T - 71iT^{2} \)
73 \( 1 + (-3.71 - 3.71i)T + 73iT^{2} \)
79 \( 1 - 10.4iT - 79T^{2} \)
83 \( 1 + (1.13 - 2.73i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (6.56 - 6.56i)T - 89iT^{2} \)
97 \( 1 - 1.01T + 97T^{2} \)
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

−9.436281721766179916597759422988, −9.117654361021907685455054319474, −8.457010238475697860877282202746, −7.72203123174491475450691109835, −6.91898973303026174533709292872, −4.97453377381626101763448130506, −4.60868553005175009713923151729, −3.71891491648526574053868580456, −2.65824176329008656263338432576, −1.26736003759065437371766346986, 1.59464464500225276120895080010, 2.80662730546739022386203173878, 3.24464481344977188891660233741, 4.32141378771114140948026007160, 6.00104613397274790007626842404, 6.95184147870198880657406023623, 7.56481251934400126319934992756, 8.219835318355180752835004597966, 8.844465258646097558403096344086, 10.06490541028187781643084031209

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