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Properties

Label 2-2e8-32.3-c2-0-10
Degree $2$
Conductor $256$
Sign $-0.107 + 0.994i$
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  + (0.299 − 0.723i)3-s + (−1.34 − 3.25i)5-s + (−0.583 − 0.583i)7-s + (5.93 + 5.93i)9-s + (−3.03 − 7.33i)11-s + (6.38 − 15.4i)13-s − 2.75·15-s − 19.0i·17-s + (−29.6 − 12.2i)19-s + (−0.596 + 0.247i)21-s + (−15.2 + 15.2i)23-s + (8.91 − 8.91i)25-s + (12.5 − 5.21i)27-s + (20.5 + 8.49i)29-s − 53.6i·31-s + ⋯
L(s)  = 1  + (0.0999 − 0.241i)3-s + (−0.269 − 0.650i)5-s + (−0.0833 − 0.0833i)7-s + (0.658 + 0.658i)9-s + (−0.276 − 0.666i)11-s + (0.491 − 1.18i)13-s − 0.183·15-s − 1.12i·17-s + (−1.56 − 0.646i)19-s + (−0.0284 + 0.0117i)21-s + (−0.665 + 0.665i)23-s + (0.356 − 0.356i)25-s + (0.466 − 0.193i)27-s + (0.707 + 0.293i)29-s − 1.73i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.107 + 0.994i$
Analytic conductor: \(6.97549\)
Root analytic conductor: \(2.64111\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1),\ -0.107 + 0.994i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.916996 - 1.02109i\)
\(L(\frac12)\) \(\approx\) \(0.916996 - 1.02109i\)
\(L(2)\) 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 \( 1 + (-0.299 + 0.723i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (1.34 + 3.25i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (0.583 + 0.583i)T + 49iT^{2} \)
11 \( 1 + (3.03 + 7.33i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (-6.38 + 15.4i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 19.0iT - 289T^{2} \)
19 \( 1 + (29.6 + 12.2i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (15.2 - 15.2i)T - 529iT^{2} \)
29 \( 1 + (-20.5 - 8.49i)T + (594. + 594. i)T^{2} \)
31 \( 1 + 53.6iT - 961T^{2} \)
37 \( 1 + (-3.80 - 9.17i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (-14.5 - 14.5i)T + 1.68e3iT^{2} \)
43 \( 1 + (-20.3 - 49.1i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 4.73T + 2.20e3T^{2} \)
53 \( 1 + (61.4 - 25.4i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-42.4 + 17.5i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-27.7 - 11.4i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (9.42 - 22.7i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-95.1 - 95.1i)T + 5.04e3iT^{2} \)
73 \( 1 + (-37.1 - 37.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 70.3T + 6.24e3T^{2} \)
83 \( 1 + (-14.5 - 6.01i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (60.8 - 60.8i)T - 7.92e3iT^{2} \)
97 \( 1 - 31.8T + 9.40e3T^{2} \)
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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

−11.48133909912360707106651455883, −10.65721623406285946530314616843, −9.627193635314982277998412248402, −8.363052422345013685891217841049, −7.85999648949247706127667186002, −6.53203121143154808382495711345, −5.27733921401926907744792654905, −4.20904994299179957805902971378, −2.60172123994831067130293733811, −0.72067071166483017342132823464, 1.92122343898173337521973111917, 3.65547435819488773151175795729, 4.48279538840447742000699813499, 6.30836874426238564961198914478, 6.87137386946018567594068690454, 8.228832918710109424001516115239, 9.172955119350045202627590074125, 10.33418650320383109417034876061, 10.82833000710720017750946077118, 12.26362498238535683075077132947

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