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Properties

Label 2-2e10-32.29-c1-0-10
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

\(L(1)\) \(\approx\) \(1.863081157\)
\(L(\frac12)\) \(\approx\) \(1.863081157\)
\(L(\frac{3}{2})\) 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

−9.711217718623727658355277835413, −9.258330490688053803120322343368, −8.702475939571200147576025114154, −7.894187465603363578476833712620, −6.53095886369046870425227222820, −6.13409964408687095145525011346, −4.61908905772106822915303305468, −3.80510835959053978921022331734, −2.93610783052837435541986209000, −1.67844176190880149081289651801, 0.77775498621428568589929139786, 2.60796032605700832401701464747, 3.08736404023408395017558553228, 4.34546058196166678387801215711, 5.40122544408152271056985297400, 6.79663653161355171053927676985, 7.20060675005214219785149818791, 8.005122754609729116362291053373, 9.075832053586103039611001226642, 9.500962535466962170755968176889

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