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Properties

Label 2-2e10-32.21-c1-0-29
Degree $2$
Conductor $1024$
Sign $-0.321 + 0.946i$
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  + (1.40 − 0.580i)3-s + (1.36 − 3.29i)5-s + (1.02 + 1.02i)7-s + (−0.492 + 0.492i)9-s + (−4.97 − 2.05i)11-s + (−1.56 − 3.78i)13-s − 5.41i·15-s − 2.35i·17-s + (−1.15 − 2.79i)19-s + (2.03 + 0.843i)21-s + (−1.06 + 1.06i)23-s + (−5.47 − 5.47i)25-s + (−2.14 + 5.18i)27-s + (3.86 − 1.60i)29-s + 10.5·31-s + ⋯
L(s)  = 1  + (0.809 − 0.335i)3-s + (0.610 − 1.47i)5-s + (0.387 + 0.387i)7-s + (−0.164 + 0.164i)9-s + (−1.49 − 0.620i)11-s + (−0.434 − 1.04i)13-s − 1.39i·15-s − 0.570i·17-s + (−0.265 − 0.641i)19-s + (0.444 + 0.183i)21-s + (−0.221 + 0.221i)23-s + (−1.09 − 1.09i)25-s + (−0.413 + 0.997i)27-s + (0.717 − 0.297i)29-s + 1.90·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-0.321 + 0.946i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ -0.321 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.954640160\)
\(L(\frac12)\) \(\approx\) \(1.954640160\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 + 0.580i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-1.36 + 3.29i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1.02 - 1.02i)T + 7iT^{2} \)
11 \( 1 + (4.97 + 2.05i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.56 + 3.78i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 2.35iT - 17T^{2} \)
19 \( 1 + (1.15 + 2.79i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (1.06 - 1.06i)T - 23iT^{2} \)
29 \( 1 + (-3.86 + 1.60i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 + (-1.75 + 4.22i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (6.27 - 6.27i)T - 41iT^{2} \)
43 \( 1 + (-8.60 - 3.56i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 3.06iT - 47T^{2} \)
53 \( 1 + (0.384 + 0.159i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (3.22 - 7.78i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-2.58 + 1.06i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-4.79 + 1.98i)T + (47.3 - 47.3i)T^{2} \)
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 \( 1 + (-3.14 - 7.60i)T + (-58.6 + 58.6i)T^{2} \)
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

−9.532424262776970295804554187570, −8.633955692865066596280352652434, −8.186965391537589683901292483498, −7.64914413219244334397493438990, −6.04766102565820641425158700986, −5.19952685734292478442036243644, −4.76678166234734332651141679262, −2.91237179112964463251454612941, −2.31018143074884540427580536328, −0.76868992471228077385462597536, 2.14714441797651357637774899999, 2.71332390964030649039199567009, 3.81413215952928839983984209138, 4.85939639136154084621491509613, 6.12930207067785758077216143008, 6.84400253712991881658843701048, 7.76780536185186271799544466755, 8.442992697039112247922973989251, 9.602491279180742941554390640708, 10.23925893707343479892218144686

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