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Properties

Label 2-2e10-32.13-c1-0-6
Degree $2$
Conductor $1024$
Sign $0.555 - 0.831i$
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  + (0.660 − 1.59i)3-s + (−0.512 + 0.212i)5-s + (−1.69 + 1.69i)7-s + (0.0177 + 0.0177i)9-s + (1.42 + 3.44i)11-s + (−5.65 − 2.34i)13-s + 0.956i·15-s + 5.26i·17-s + (3.80 + 1.57i)19-s + (1.58 + 3.82i)21-s + (4.31 + 4.31i)23-s + (−3.31 + 3.31i)25-s + (4.82 − 1.99i)27-s + (−0.512 + 1.23i)29-s − 1.53·31-s + ⋯
L(s)  = 1  + (0.381 − 0.920i)3-s + (−0.229 + 0.0949i)5-s + (−0.642 + 0.642i)7-s + (0.00590 + 0.00590i)9-s + (0.429 + 1.03i)11-s + (−1.56 − 0.649i)13-s + 0.247i·15-s + 1.27i·17-s + (0.871 + 0.361i)19-s + (0.346 + 0.835i)21-s + (0.899 + 0.899i)23-s + (−0.663 + 0.663i)25-s + (0.927 − 0.384i)27-s + (−0.0951 + 0.229i)29-s − 0.274·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $0.555 - 0.831i$
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: \((2,\ 1024,\ (\ :1/2),\ 0.555 - 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.307059411\)
\(L(\frac12)\) \(\approx\) \(1.307059411\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.660 + 1.59i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.512 - 0.212i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.69 - 1.69i)T - 7iT^{2} \)
11 \( 1 + (-1.42 - 3.44i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (5.65 + 2.34i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 5.26iT - 17T^{2} \)
19 \( 1 + (-3.80 - 1.57i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-4.31 - 4.31i)T + 23iT^{2} \)
29 \( 1 + (0.512 - 1.23i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 1.53T + 31T^{2} \)
37 \( 1 + (-3.61 + 1.49i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (8.69 + 8.69i)T + 41iT^{2} \)
43 \( 1 + (0.211 + 0.511i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 9.73iT - 47T^{2} \)
53 \( 1 + (-2.78 - 6.73i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-2.76 + 1.14i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-3.06 + 7.40i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (2.20 - 5.33i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-9.45 + 9.45i)T - 71iT^{2} \)
73 \( 1 + (1.69 + 1.69i)T + 73iT^{2} \)
79 \( 1 - 12.8iT - 79T^{2} \)
83 \( 1 + (0.870 + 0.360i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (1.14 - 1.14i)T - 89iT^{2} \)
97 \( 1 + 1.83T + 97T^{2} \)
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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

−9.812280642000373696001484325767, −9.391994987874083000808931594215, −8.218112755919638927600948337086, −7.39803025701183180504527548942, −7.06430124989728747983008485047, −5.86813699667514983477883457014, −4.96219916318560128886156190931, −3.62519743953937409993528082181, −2.55281757228585084746678037347, −1.57155625181423562233948339650, 0.57175287834641380725677298103, 2.71452767940072362524288427869, 3.52898025978820727817184647374, 4.47313488387792517893465229402, 5.18902515036803629796041505186, 6.68591169457915848737324130177, 7.13371676526085576691012186558, 8.338494357763024768681611864037, 9.239685768670713838633414771247, 9.756426311758503802255024446935

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