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Properties

Label 2-2e10-32.13-c1-0-9
Degree $2$
Conductor $1024$
Sign $0.997 + 0.0654i$
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

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.334247180\)
\(L(\frac12)\) \(\approx\) \(1.334247180\)
\(L(\frac{3}{2})\) 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

−10.04781181109904322385666501908, −8.687257830996159579530880757444, −8.129105070375911236283798840549, −7.67055747415890972871783095683, −6.57753788195940520009035474286, −5.99512332566014060871677019781, −4.47517008766522573194491480474, −3.40863576683559628614986092092, −2.75853591624871907522810901768, −0.987721438073612037700659298721, 0.824889174152015938381849805190, 2.91716490433607777852822298417, 3.89725360277816594972215956873, 4.43906463448743910835251572485, 5.28705761346352511103351663232, 6.92443404152974384306467700402, 7.43026372367016114073011328634, 8.435954440488364979377247377097, 9.045465242499456668415447398565, 10.04709504730951797829884841950

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