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Properties

Label 2-2e8-32.11-c2-0-8
Degree $2$
Conductor $256$
Sign $-0.201 + 0.979i$
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  + (−1.10 − 2.67i)3-s + (−2.95 + 7.13i)5-s + (4.18 − 4.18i)7-s + (0.437 − 0.437i)9-s + (1.42 − 3.44i)11-s + (−8.39 − 20.2i)13-s + 22.3·15-s − 1.73i·17-s + (14.2 − 5.90i)19-s + (−15.8 − 6.55i)21-s + (−15.1 − 15.1i)23-s + (−24.5 − 24.5i)25-s + (−25.7 − 10.6i)27-s + (6.74 − 2.79i)29-s − 31.1i·31-s + ⋯
L(s)  = 1  + (−0.369 − 0.891i)3-s + (−0.591 + 1.42i)5-s + (0.597 − 0.597i)7-s + (0.0486 − 0.0486i)9-s + (0.129 − 0.313i)11-s + (−0.646 − 1.55i)13-s + 1.49·15-s − 0.101i·17-s + (0.749 − 0.310i)19-s + (−0.753 − 0.312i)21-s + (−0.658 − 0.658i)23-s + (−0.980 − 0.980i)25-s + (−0.952 − 0.394i)27-s + (0.232 − 0.0962i)29-s − 1.00i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.201 + 0.979i$
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: \((2,\ 256,\ (\ :1),\ -0.201 + 0.979i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.693378 - 0.850139i\)
\(L(\frac12)\) \(\approx\) \(0.693378 - 0.850139i\)
\(L(2)\) 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 \( 1 + (1.10 + 2.67i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (2.95 - 7.13i)T + (-17.6 - 17.6i)T^{2} \)
7 \( 1 + (-4.18 + 4.18i)T - 49iT^{2} \)
11 \( 1 + (-1.42 + 3.44i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (8.39 + 20.2i)T + (-119. + 119. i)T^{2} \)
17 \( 1 + 1.73iT - 289T^{2} \)
19 \( 1 + (-14.2 + 5.90i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (15.1 + 15.1i)T + 529iT^{2} \)
29 \( 1 + (-6.74 + 2.79i)T + (594. - 594. i)T^{2} \)
31 \( 1 + 31.1iT - 961T^{2} \)
37 \( 1 + (5.30 - 12.7i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (18.5 - 18.5i)T - 1.68e3iT^{2} \)
43 \( 1 + (-31.0 + 75.0i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 16.2T + 2.20e3T^{2} \)
53 \( 1 + (-29.0 - 12.0i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-34.1 - 14.1i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-68.7 + 28.4i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-10.5 - 25.3i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-32.2 + 32.2i)T - 5.04e3iT^{2} \)
73 \( 1 + (28.5 - 28.5i)T - 5.32e3iT^{2} \)
79 \( 1 + 22.4T + 6.24e3T^{2} \)
83 \( 1 + (123. - 51.0i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-61.0 - 61.0i)T + 7.92e3iT^{2} \)
97 \( 1 + 69.9T + 9.40e3T^{2} \)
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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

−11.55144996527415334729852984802, −10.70929127458743839933167219788, −9.954355778307568249375908157884, −8.101639295120862808270251929191, −7.44614767027844797756111909949, −6.77677891704086123459350673576, −5.59193766167766413026113714013, −3.91201057158287358174989601185, −2.61536827907804452044475705387, −0.62055836622506919302740020324, 1.68521809831610264964249902185, 4.03977889324213494057377461759, 4.75389032919000365029815850117, 5.49774782480212926286310197774, 7.25640813639863401779531497185, 8.385979369414449209492311552737, 9.232974054182386369970331683889, 9.961056324258733034219097698202, 11.39968164646300752066757881651, 11.88242228405673523536284365362

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