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Properties

Label 2-2e8-32.11-c2-0-13
Degree $2$
Conductor $256$
Sign $-0.883 + 0.468i$
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  + (−1.73 − 4.18i)3-s + (1.85 − 4.48i)5-s + (5.27 − 5.27i)7-s + (−8.12 + 8.12i)9-s + (6.20 − 14.9i)11-s + (4.22 + 10.2i)13-s − 21.9·15-s + 2.84i·17-s + (−12.4 + 5.14i)19-s + (−31.2 − 12.9i)21-s + (1.43 + 1.43i)23-s + (0.999 + 0.999i)25-s + (10.4 + 4.30i)27-s + (−36.9 + 15.3i)29-s − 4.73i·31-s + ⋯
L(s)  = 1  + (−0.577 − 1.39i)3-s + (0.371 − 0.897i)5-s + (0.753 − 0.753i)7-s + (−0.902 + 0.902i)9-s + (0.563 − 1.36i)11-s + (0.325 + 0.784i)13-s − 1.46·15-s + 0.167i·17-s + (−0.654 + 0.270i)19-s + (−1.48 − 0.615i)21-s + (0.0625 + 0.0625i)23-s + (0.0399 + 0.0399i)25-s + (0.385 + 0.159i)27-s + (−1.27 + 0.527i)29-s − 0.152i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.883 + 0.468i$
Analytic conductor: \(6.97549\)
Root analytic conductor: \(2.64111\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1),\ -0.883 + 0.468i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.341272 - 1.37146i\)
\(L(\frac12)\) \(\approx\) \(0.341272 - 1.37146i\)
\(L(2)\) 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 \( 1 + (1.73 + 4.18i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (-1.85 + 4.48i)T + (-17.6 - 17.6i)T^{2} \)
7 \( 1 + (-5.27 + 5.27i)T - 49iT^{2} \)
11 \( 1 + (-6.20 + 14.9i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (-4.22 - 10.2i)T + (-119. + 119. i)T^{2} \)
17 \( 1 - 2.84iT - 289T^{2} \)
19 \( 1 + (12.4 - 5.14i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (-1.43 - 1.43i)T + 529iT^{2} \)
29 \( 1 + (36.9 - 15.3i)T + (594. - 594. i)T^{2} \)
31 \( 1 + 4.73iT - 961T^{2} \)
37 \( 1 + (-6.68 + 16.1i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (-40.4 + 40.4i)T - 1.68e3iT^{2} \)
43 \( 1 + (24.5 - 59.1i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 16.5T + 2.20e3T^{2} \)
53 \( 1 + (-46.9 - 19.4i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-50.0 - 20.7i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-54.3 + 22.4i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (25.5 + 61.5i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (7.12 - 7.12i)T - 5.04e3iT^{2} \)
73 \( 1 + (-55.3 + 55.3i)T - 5.32e3iT^{2} \)
79 \( 1 + 11.0T + 6.24e3T^{2} \)
83 \( 1 + (29.9 - 12.4i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-16.7 - 16.7i)T + 7.92e3iT^{2} \)
97 \( 1 + 67.8T + 9.40e3T^{2} \)
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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

−11.39940602907004348721148155248, −10.90912480380546976375132004360, −9.191083051163547987455617816834, −8.354190051599480271524217969705, −7.36941134431353651608981636539, −6.32314168358689892341206144462, −5.47881211930667049322680893654, −4.04209789926374440366051446252, −1.73763889288102759041080922611, −0.843585980819484012768857764389, 2.32197971583464122367538149964, 3.92083210014074230388710780897, 4.98040909085577531958556582607, 5.88607239060938135274644467710, 7.10184907990153577556954731234, 8.560871377264332064158563292205, 9.641561028802183740110607515446, 10.27688498182950978976645379124, 11.11947588147376474521266840521, 11.81776952660833132213184134544

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