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Properties

Label 2-2e10-32.21-c1-0-10
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

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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: $\chi_{1024} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ 0.980 - 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5156392966\)
\(L(\frac12)\) \(\approx\) \(0.5156392966\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.01T + 97T^{2} \)
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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

−10.31718066040305996364733316309, −9.609748944101349679305216900447, −8.163994194134011029337499446207, −7.04465418431688141662092497347, −6.60501670733120334604242063922, −5.73709758039463554843161930273, −4.76679078664398692076393391433, −3.87323445334885823489023929993, −2.83713675115428956141418758210, −0.46498324645547429122342443271, 0.75370002953644645735345914068, 2.00213776224797870580014442832, 4.08816332939433883354662796831, 4.89263559525341508597456623510, 5.58642477219226919990061206631, 6.41923856171519021575807001876, 7.27549228094781588761081459998, 8.157142185624375740709577201881, 9.034369762122264942788596621164, 10.11470044590968482008757418026

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