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Properties

Label 2-2e10-32.29-c1-0-2
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.59 − 0.660i)3-s + (−0.212 − 0.512i)5-s + (1.69 − 1.69i)7-s + (−0.0177 − 0.0177i)9-s + (−3.44 + 1.42i)11-s + (−2.34 + 5.65i)13-s + 0.956i·15-s + 5.26i·17-s + (1.57 − 3.80i)19-s + (−3.82 + 1.58i)21-s + (−4.31 − 4.31i)23-s + (3.31 − 3.31i)25-s + (1.99 + 4.82i)27-s + (1.23 + 0.512i)29-s − 1.53·31-s + ⋯
L(s)  = 1  + (−0.920 − 0.381i)3-s + (−0.0949 − 0.229i)5-s + (0.642 − 0.642i)7-s + (−0.00590 − 0.00590i)9-s + (−1.03 + 0.429i)11-s + (−0.649 + 1.56i)13-s + 0.247i·15-s + 1.27i·17-s + (0.361 − 0.871i)19-s + (−0.835 + 0.346i)21-s + (−0.899 − 0.899i)23-s + (0.663 − 0.663i)25-s + (0.384 + 0.927i)27-s + (0.229 + 0.0951i)29-s − 0.274·31-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: $\chi_{1024} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ 0.195 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5963463508\)
\(L(\frac12)\) \(\approx\) \(0.5963463508\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 + 0.660i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (0.212 + 0.512i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.69 + 1.69i)T - 7iT^{2} \)
11 \( 1 + (3.44 - 1.42i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (2.34 - 5.65i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 5.26iT - 17T^{2} \)
19 \( 1 + (-1.57 + 3.80i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (4.31 + 4.31i)T + 23iT^{2} \)
29 \( 1 + (-1.23 - 0.512i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 1.53T + 31T^{2} \)
37 \( 1 + (-1.49 - 3.61i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-8.69 - 8.69i)T + 41iT^{2} \)
43 \( 1 + (-0.511 + 0.211i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 9.73iT - 47T^{2} \)
53 \( 1 + (6.73 - 2.78i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.14 - 2.76i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (7.40 + 3.06i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-5.33 - 2.20i)T + (47.3 + 47.3i)T^{2} \)
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 \( 1 + (0.360 - 0.870i)T + (-58.6 - 58.6i)T^{2} \)
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

−10.31652965470163454511158975249, −9.353463517480823487748745720971, −8.330262099810082104750227407267, −7.54180398408535115732325979527, −6.68819647400356700333553167361, −5.97458558880843867323162084537, −4.70097335929552900937032306490, −4.41068719288114514210047085000, −2.58206930411908315145112141071, −1.25717925560309613086482086409, 0.32540050011981413906178336287, 2.35269146799655787619811597282, 3.35984876501527087907563469262, 4.95765485523655012712565553886, 5.37550790705034415574638183603, 5.93752116799537346043386671287, 7.54025148248032544853132903429, 7.84593720280132102739256423915, 8.977089186291229145933888179802, 10.06702018310218689965581815254

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