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Properties

Label 2-2e8-32.19-c2-0-12
Degree $2$
Conductor $256$
Sign $-0.936 + 0.351i$
Analytic cond. $6.97549$
Root an. cond. $2.64111$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.58 + 0.655i)3-s + (−4.18 + 1.73i)5-s + (−3.93 − 3.93i)7-s + (−4.29 − 4.29i)9-s + (−14.2 + 5.89i)11-s + (−0.454 − 0.188i)13-s − 7.76·15-s − 26.5i·17-s + (−7.25 + 17.5i)19-s + (−3.64 − 8.79i)21-s + (−0.775 + 0.775i)23-s + (−3.14 + 3.14i)25-s + (−9.87 − 23.8i)27-s + (17.9 − 43.4i)29-s + 39.6i·31-s + ⋯
L(s)  = 1  + (0.527 + 0.218i)3-s + (−0.837 + 0.346i)5-s + (−0.561 − 0.561i)7-s + (−0.476 − 0.476i)9-s + (−1.29 + 0.536i)11-s + (−0.0349 − 0.0144i)13-s − 0.517·15-s − 1.56i·17-s + (−0.381 + 0.921i)19-s + (−0.173 − 0.418i)21-s + (−0.0337 + 0.0337i)23-s + (−0.125 + 0.125i)25-s + (−0.365 − 0.882i)27-s + (0.620 − 1.49i)29-s + 1.28i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.936 + 0.351i$
Analytic conductor: \(6.97549\)
Root analytic conductor: \(2.64111\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1),\ -0.936 + 0.351i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0298981 - 0.164872i\)
\(L(\frac12)\) \(\approx\) \(0.0298981 - 0.164872i\)
\(L(2)\) 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 \( 1 + (-1.58 - 0.655i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (4.18 - 1.73i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (3.93 + 3.93i)T + 49iT^{2} \)
11 \( 1 + (14.2 - 5.89i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (0.454 + 0.188i)T + (119. + 119. i)T^{2} \)
17 \( 1 + 26.5iT - 289T^{2} \)
19 \( 1 + (7.25 - 17.5i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (0.775 - 0.775i)T - 529iT^{2} \)
29 \( 1 + (-17.9 + 43.4i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 39.6iT - 961T^{2} \)
37 \( 1 + (36.4 - 15.1i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-38.9 - 38.9i)T + 1.68e3iT^{2} \)
43 \( 1 + (14.2 - 5.91i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 62.1T + 2.20e3T^{2} \)
53 \( 1 + (11.4 + 27.7i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (5.30 + 12.8i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (14.1 - 34.1i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-26.1 - 10.8i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (17.7 + 17.7i)T + 5.04e3iT^{2} \)
73 \( 1 + (-12.8 - 12.8i)T + 5.32e3iT^{2} \)
79 \( 1 - 144.T + 6.24e3T^{2} \)
83 \( 1 + (10.9 - 26.5i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (5.92 - 5.92i)T - 7.92e3iT^{2} \)
97 \( 1 - 66.9T + 9.40e3T^{2} \)
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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

−11.42451332258343747865965199776, −10.25397560644528756130079553107, −9.607000158056878826363482763190, −8.274157607227136276225325131947, −7.56435960560762170699674372711, −6.50634860925370306402495743350, −4.96039721483125242784000466489, −3.66225019713450203020991056704, −2.74725862416625552978725259952, −0.07523234752416594368377212072, 2.37420887238025865134979554260, 3.52436108539597649274705205261, 5.01777253524314312531752141090, 6.14483082676240182576725603506, 7.60774230634375513261927344751, 8.340820403812617984450764297331, 8.983065774682442980284628794016, 10.47519506220582659400412833307, 11.18485019551317165913531457571, 12.46642797974340544168191858412

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