Properties

Label 2-2e9-512.141-c1-0-29
Degree $2$
Conductor $512$
Sign $0.785 - 0.618i$
Analytic cond. $4.08834$
Root an. cond. $2.02196$
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  + (1.33 − 0.467i)2-s + (0.235 − 0.713i)3-s + (1.56 − 1.24i)4-s + (−1.59 + 4.12i)5-s + (−0.0193 − 1.06i)6-s + (−1.92 + 3.20i)7-s + (1.50 − 2.39i)8-s + (1.95 + 1.45i)9-s + (−0.193 + 6.25i)10-s + (0.281 + 2.28i)11-s + (−0.522 − 1.40i)12-s + (0.0983 − 4.00i)13-s + (−1.06 + 5.18i)14-s + (2.56 + 2.10i)15-s + (0.880 − 3.90i)16-s + (−0.616 − 0.751i)17-s + ⋯
L(s)  = 1  + (0.943 − 0.330i)2-s + (0.136 − 0.411i)3-s + (0.781 − 0.624i)4-s + (−0.712 + 1.84i)5-s + (−0.00788 − 0.433i)6-s + (−0.727 + 1.21i)7-s + (0.530 − 0.847i)8-s + (0.652 + 0.483i)9-s + (−0.0611 + 1.97i)10-s + (0.0848 + 0.687i)11-s + (−0.150 − 0.406i)12-s + (0.0272 − 1.11i)13-s + (−0.284 + 1.38i)14-s + (0.663 + 0.544i)15-s + (0.220 − 0.975i)16-s + (−0.149 − 0.182i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 - 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.785 - 0.618i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.785 - 0.618i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14046 + 0.741223i\)
\(L(\frac12)\) \(\approx\) \(2.14046 + 0.741223i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 + 0.467i)T \)
good3 \( 1 + (-0.235 + 0.713i)T + (-2.40 - 1.78i)T^{2} \)
5 \( 1 + (1.59 - 4.12i)T + (-3.70 - 3.35i)T^{2} \)
7 \( 1 + (1.92 - 3.20i)T + (-3.29 - 6.17i)T^{2} \)
11 \( 1 + (-0.281 - 2.28i)T + (-10.6 + 2.67i)T^{2} \)
13 \( 1 + (-0.0983 + 4.00i)T + (-12.9 - 0.637i)T^{2} \)
17 \( 1 + (0.616 + 0.751i)T + (-3.31 + 16.6i)T^{2} \)
19 \( 1 + (-4.27 - 3.01i)T + (6.40 + 17.8i)T^{2} \)
23 \( 1 + (1.12 - 2.37i)T + (-14.5 - 17.7i)T^{2} \)
29 \( 1 + (7.78 - 2.15i)T + (24.8 - 14.9i)T^{2} \)
31 \( 1 + (-9.73 + 1.93i)T + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (0.341 + 1.52i)T + (-33.4 + 15.8i)T^{2} \)
41 \( 1 + (-8.55 + 7.74i)T + (4.01 - 40.8i)T^{2} \)
43 \( 1 + (-1.50 + 2.99i)T + (-25.6 - 34.5i)T^{2} \)
47 \( 1 + (-2.48 - 1.33i)T + (26.1 + 39.0i)T^{2} \)
53 \( 1 + (3.44 + 0.954i)T + (45.4 + 27.2i)T^{2} \)
59 \( 1 + (-2.08 + 0.0511i)T + (58.9 - 2.89i)T^{2} \)
61 \( 1 + (2.90 + 2.50i)T + (8.95 + 60.3i)T^{2} \)
67 \( 1 + (3.43 - 0.253i)T + (66.2 - 9.83i)T^{2} \)
71 \( 1 + (1.20 - 8.09i)T + (-67.9 - 20.6i)T^{2} \)
73 \( 1 + (2.79 + 4.66i)T + (-34.4 + 64.3i)T^{2} \)
79 \( 1 + (-10.9 - 3.33i)T + (65.6 + 43.8i)T^{2} \)
83 \( 1 + (0.983 - 4.37i)T + (-75.0 - 35.4i)T^{2} \)
89 \( 1 + (2.66 + 5.62i)T + (-56.4 + 68.7i)T^{2} \)
97 \( 1 + (-1.39 - 0.931i)T + (37.1 + 89.6i)T^{2} \)
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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

−11.11558753865269537157384363223, −10.29840174816615926929188073395, −9.650372277126912359780917582121, −7.74768347967067264573241557221, −7.30557741881758777263747670840, −6.33123797594237592401859384570, −5.52151703352023493358033628237, −3.93475208490501396968808685904, −2.98570606845762614271024549373, −2.25664605454196992476088640720, 1.06709224851424264808916010610, 3.44760457664012043251353849862, 4.28458348798008988831016237737, 4.65697182421495959463331168179, 6.08206519647094791951837144073, 7.11579839899728138725311297770, 7.974812575366322780254517911320, 9.025978572248067206184267711157, 9.734189796761310109066243222079, 11.10649854794157834225386181777

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