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Properties

Label 2-2e8-32.5-c1-0-0
Degree $2$
Conductor $256$
Sign $-0.0493 - 0.998i$
Analytic cond. $2.04417$
Root an. cond. $1.42974$
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  + (1.07 + 2.60i)3-s + (−0.707 − 0.292i)5-s + (1.68 + 1.68i)7-s + (−3.50 + 3.50i)9-s + (0.334 − 0.808i)11-s + (−1.09 + 0.451i)13-s − 2.15i·15-s − 0.224i·17-s + (2.87 − 1.19i)19-s + (−2.57 + 6.21i)21-s + (−3.68 + 3.68i)23-s + (−3.12 − 3.12i)25-s + (−5.09 − 2.11i)27-s + (−2.34 − 5.66i)29-s + 6.82·31-s + ⋯
L(s)  = 1  + (0.623 + 1.50i)3-s + (−0.316 − 0.130i)5-s + (0.637 + 0.637i)7-s + (−1.16 + 1.16i)9-s + (0.100 − 0.243i)11-s + (−0.302 + 0.125i)13-s − 0.557i·15-s − 0.0545i·17-s + (0.660 − 0.273i)19-s + (−0.561 + 1.35i)21-s + (−0.768 + 0.768i)23-s + (−0.624 − 0.624i)25-s + (−0.981 − 0.406i)27-s + (−0.435 − 1.05i)29-s + 1.22·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0493 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0493 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.0493 - 0.998i$
Analytic conductor: \(2.04417\)
Root analytic conductor: \(1.42974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1/2),\ -0.0493 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01662 + 1.06810i\)
\(L(\frac12)\) \(\approx\) \(1.01662 + 1.06810i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.07 - 2.60i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (0.707 + 0.292i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.68 - 1.68i)T + 7iT^{2} \)
11 \( 1 + (-0.334 + 0.808i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.09 - 0.451i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 0.224iT - 17T^{2} \)
19 \( 1 + (-2.87 + 1.19i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (3.68 - 3.68i)T - 23iT^{2} \)
29 \( 1 + (2.34 + 5.66i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 6.82T + 31T^{2} \)
37 \( 1 + (-9.87 - 4.09i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-6.37 + 6.37i)T - 41iT^{2} \)
43 \( 1 + (1.90 - 4.60i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 0.542iT - 47T^{2} \)
53 \( 1 + (-3.91 + 9.46i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-3.36 - 1.39i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (0.398 + 0.962i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (1.48 + 3.57i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (5.39 + 5.39i)T + 71iT^{2} \)
73 \( 1 + (5.15 - 5.15i)T - 73iT^{2} \)
79 \( 1 + 8.39iT - 79T^{2} \)
83 \( 1 + (11.2 - 4.64i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.92 + 5.92i)T + 89iT^{2} \)
97 \( 1 + 4.19T + 97T^{2} \)
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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

−11.83667123775791197787237124125, −11.39490885297362965942307734758, −10.08234411663248923598795592985, −9.507466521248476465485879863858, −8.499064637928379381466355838907, −7.77742912331823583556271938037, −5.88703175555853376478671470014, −4.78378678125920623413775664213, −3.91185533732303111879007167076, −2.55947028632143661203284302200, 1.27510531847421322742468106201, 2.70046719743264773533628998816, 4.25381588494014302788364236237, 5.93633687391509359034291292222, 7.17607318889180073043625774610, 7.67066437340506574720410677322, 8.488677966193811377510544343802, 9.772088526431784120147726099282, 11.05920495738220043653086924425, 11.96917732185858104420208624736

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