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Properties

Label 2-2e10-32.5-c1-0-11
Degree $2$
Conductor $1024$
Sign $-0.0654 - 0.997i$
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  + (0.580 + 1.40i)3-s + (3.29 + 1.36i)5-s + (1.02 + 1.02i)7-s + (0.492 − 0.492i)9-s + (−2.05 + 4.97i)11-s + (−3.78 + 1.56i)13-s + 5.41i·15-s − 2.35i·17-s + (2.79 − 1.15i)19-s + (−0.843 + 2.03i)21-s + (−1.06 + 1.06i)23-s + (5.47 + 5.47i)25-s + (5.18 + 2.14i)27-s + (−1.60 − 3.86i)29-s − 10.5·31-s + ⋯
L(s)  = 1  + (0.335 + 0.809i)3-s + (1.47 + 0.610i)5-s + (0.387 + 0.387i)7-s + (0.164 − 0.164i)9-s + (−0.620 + 1.49i)11-s + (−1.04 + 0.434i)13-s + 1.39i·15-s − 0.570i·17-s + (0.641 − 0.265i)19-s + (−0.183 + 0.444i)21-s + (−0.221 + 0.221i)23-s + (1.09 + 1.09i)25-s + (0.997 + 0.413i)27-s + (−0.297 − 0.717i)29-s − 1.90·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0654 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0654 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-0.0654 - 0.997i$
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: \((2,\ 1024,\ (\ :1/2),\ -0.0654 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.298319382\)
\(L(\frac12)\) \(\approx\) \(2.298319382\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.580 - 1.40i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-3.29 - 1.36i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.02 - 1.02i)T + 7iT^{2} \)
11 \( 1 + (2.05 - 4.97i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (3.78 - 1.56i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 2.35iT - 17T^{2} \)
19 \( 1 + (-2.79 + 1.15i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.06 - 1.06i)T - 23iT^{2} \)
29 \( 1 + (1.60 + 3.86i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 10.5T + 31T^{2} \)
37 \( 1 + (-4.22 - 1.75i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-6.27 + 6.27i)T - 41iT^{2} \)
43 \( 1 + (-3.56 + 8.60i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 3.06iT - 47T^{2} \)
53 \( 1 + (-0.159 + 0.384i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-7.78 - 3.22i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.06 + 2.58i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-1.98 - 4.79i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-2.84 - 2.84i)T + 71iT^{2} \)
73 \( 1 + (8.43 - 8.43i)T - 73iT^{2} \)
79 \( 1 - 4.59iT - 79T^{2} \)
83 \( 1 + (7.60 - 3.14i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-0.967 - 0.967i)T + 89iT^{2} \)
97 \( 1 - 11.2T + 97T^{2} \)
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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

−9.858808876012021872218665146464, −9.632340918792888128316862014947, −8.928996097741546866531301724773, −7.36618053069979439497605166706, −7.01490409526902534673287783408, −5.60055786386829808991852323609, −5.07488417293295318083526661534, −4.01745202072048026675284518802, −2.55737653717288451985140856952, −2.02445798664020420460706641565, 1.04418197011229831327672526192, 2.02864981988841698639582596975, 3.04335513283378722333499850701, 4.67752318644764968852533630056, 5.57801223078008893610025016935, 6.14533329133472959195145916462, 7.45314023172418309580351412727, 7.936463407580019901354684172549, 8.892046711032029651453590221283, 9.658951064561942208423792575919

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