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Properties

Label 2-2e10-32.13-c1-0-1
Degree $2$
Conductor $1024$
Sign $-0.659 + 0.751i$
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  + (−1.12 + 2.70i)3-s + (0.366 − 0.151i)5-s + (−3.06 + 3.06i)7-s + (−3.95 − 3.95i)9-s + (1.51 + 3.66i)11-s + (1.88 + 0.780i)13-s + 1.16i·15-s + 4.54i·17-s + (0.534 + 0.221i)19-s + (−4.86 − 11.7i)21-s + (−4.41 − 4.41i)23-s + (−3.42 + 3.42i)25-s + (7.03 − 2.91i)27-s + (1.96 − 4.74i)29-s + 0.0539·31-s + ⋯
L(s)  = 1  + (−0.647 + 1.56i)3-s + (0.163 − 0.0678i)5-s + (−1.15 + 1.15i)7-s + (−1.31 − 1.31i)9-s + (0.457 + 1.10i)11-s + (0.522 + 0.216i)13-s + 0.299i·15-s + 1.10i·17-s + (0.122 + 0.0508i)19-s + (−1.06 − 2.56i)21-s + (−0.921 − 0.921i)23-s + (−0.684 + 0.684i)25-s + (1.35 − 0.560i)27-s + (0.365 − 0.881i)29-s + 0.00969·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-0.659 + 0.751i$
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: \((2,\ 1024,\ (\ :1/2),\ -0.659 + 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6460600411\)
\(L(\frac12)\) \(\approx\) \(0.6460600411\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.12 - 2.70i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.366 + 0.151i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (3.06 - 3.06i)T - 7iT^{2} \)
11 \( 1 + (-1.51 - 3.66i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-1.88 - 0.780i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 4.54iT - 17T^{2} \)
19 \( 1 + (-0.534 - 0.221i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (4.41 + 4.41i)T + 23iT^{2} \)
29 \( 1 + (-1.96 + 4.74i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 0.0539T + 31T^{2} \)
37 \( 1 + (0.798 - 0.330i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.621 + 0.621i)T + 41iT^{2} \)
43 \( 1 + (0.857 + 2.06i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 9.44iT - 47T^{2} \)
53 \( 1 + (-4.16 - 10.0i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-7.17 + 2.97i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-4.02 + 9.72i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (3.12 - 7.53i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-2.99 + 2.99i)T - 71iT^{2} \)
73 \( 1 + (-2.91 - 2.91i)T + 73iT^{2} \)
79 \( 1 - 5.74iT - 79T^{2} \)
83 \( 1 + (3.30 + 1.36i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (2.38 - 2.38i)T - 89iT^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−10.09492059058471165212582127158, −9.874336755890089356217851072588, −9.097262830160879836441520880187, −8.369160828174766394228450287679, −6.71872909948606720671491116404, −6.01456494265842337495655425586, −5.39082262886680056579874345189, −4.22048691380754459050043034264, −3.64218662229003838254548424660, −2.23963033972918804271834667002, 0.33868912465268725348260795809, 1.29234809107580018563708963374, 2.87195440349871545071912762083, 3.87575364087413221418571293585, 5.49179250815963191473516496572, 6.23634214671830042556702175752, 6.79557160766644315303119008891, 7.50572861100167512566810089912, 8.339277422333178945632249787665, 9.479170378475868472428569940976

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