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Properties

Label 2-2e10-32.5-c1-0-17
Degree $2$
Conductor $1024$
Sign $0.946 + 0.321i$
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  + (−0.184 − 0.445i)3-s + (1.36 + 0.565i)5-s + (−0.135 − 0.135i)7-s + (1.95 − 1.95i)9-s + (−1.29 + 3.12i)11-s + (2.29 − 0.951i)13-s − 0.713i·15-s − 3.11i·17-s + (5.99 − 2.48i)19-s + (−0.0352 + 0.0851i)21-s + (−5.18 + 5.18i)23-s + (−1.98 − 1.98i)25-s + (−2.57 − 1.06i)27-s + (1.79 + 4.33i)29-s + 7.44·31-s + ⋯
L(s)  = 1  + (−0.106 − 0.257i)3-s + (0.610 + 0.253i)5-s + (−0.0510 − 0.0510i)7-s + (0.652 − 0.652i)9-s + (−0.390 + 0.941i)11-s + (0.637 − 0.263i)13-s − 0.184i·15-s − 0.754i·17-s + (1.37 − 0.569i)19-s + (−0.00769 + 0.0185i)21-s + (−1.08 + 1.08i)23-s + (−0.397 − 0.397i)25-s + (−0.494 − 0.204i)27-s + (0.333 + 0.804i)29-s + 1.33·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $0.946 + 0.321i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ 0.946 + 0.321i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.851207071\)
\(L(\frac12)\) \(\approx\) \(1.851207071\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.184 + 0.445i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.36 - 0.565i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.135 + 0.135i)T + 7iT^{2} \)
11 \( 1 + (1.29 - 3.12i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-2.29 + 0.951i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 3.11iT - 17T^{2} \)
19 \( 1 + (-5.99 + 2.48i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (5.18 - 5.18i)T - 23iT^{2} \)
29 \( 1 + (-1.79 - 4.33i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 7.44T + 31T^{2} \)
37 \( 1 + (-8.44 - 3.49i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-4.27 + 4.27i)T - 41iT^{2} \)
43 \( 1 + (1.79 - 4.33i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 12.0iT - 47T^{2} \)
53 \( 1 + (1.41 - 3.42i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (2.81 + 1.16i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-3.61 - 8.72i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (2.96 + 7.15i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-2.86 - 2.86i)T + 71iT^{2} \)
73 \( 1 + (-2.49 + 2.49i)T - 73iT^{2} \)
79 \( 1 - 8.39iT - 79T^{2} \)
83 \( 1 + (-13.0 + 5.42i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-4.96 - 4.96i)T + 89iT^{2} \)
97 \( 1 + 2.87T + 97T^{2} \)
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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

−9.759645370862398163550589376487, −9.440373607281307343858408737518, −8.090711213907960779856589667855, −7.30077690011427645984289226375, −6.57859778770152284405311469703, −5.68635206949372450795310073280, −4.71997450558963764024520245407, −3.55532455533006862150055961315, −2.38395827150623597082700003431, −1.09002967874563692015262484591, 1.23189174058111699002930815077, 2.52654053749706861115801020778, 3.83725392471474936621195472567, 4.74287735010137971941805626417, 5.87648924375007971461811310926, 6.23671362206796225681031341656, 7.79090082286863357900643954261, 8.159902902527990833970758264242, 9.361713984264147370959997000347, 9.949588460493641276937043234486

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