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Properties

Label 2-2e8-32.19-c2-0-6
Degree $2$
Conductor $256$
Sign $0.974 - 0.224i$
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  + (2.49 + 1.03i)3-s + (0.452 − 0.187i)5-s + (−0.429 − 0.429i)7-s + (−1.19 − 1.19i)9-s + (17.3 − 7.18i)11-s + (19.9 + 8.26i)13-s + 1.32·15-s + 13.5i·17-s + (−3.45 + 8.34i)19-s + (−0.628 − 1.51i)21-s + (16.8 − 16.8i)23-s + (−17.5 + 17.5i)25-s + (−11.0 − 26.7i)27-s + (−13.8 + 33.4i)29-s − 24.5i·31-s + ⋯
L(s)  = 1  + (0.832 + 0.344i)3-s + (0.0904 − 0.0374i)5-s + (−0.0614 − 0.0614i)7-s + (−0.133 − 0.133i)9-s + (1.57 − 0.652i)11-s + (1.53 + 0.635i)13-s + 0.0882·15-s + 0.799i·17-s + (−0.182 + 0.439i)19-s + (−0.0299 − 0.0722i)21-s + (0.734 − 0.734i)23-s + (−0.700 + 0.700i)25-s + (−0.409 − 0.989i)27-s + (−0.477 + 1.15i)29-s − 0.792i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.224i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.974 - 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.974 - 0.224i$
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: \((2,\ 256,\ (\ :1),\ 0.974 - 0.224i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.23548 + 0.253875i\)
\(L(\frac12)\) \(\approx\) \(2.23548 + 0.253875i\)
\(L(2)\) 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 \( 1 + (-2.49 - 1.03i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (-0.452 + 0.187i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (0.429 + 0.429i)T + 49iT^{2} \)
11 \( 1 + (-17.3 + 7.18i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (-19.9 - 8.26i)T + (119. + 119. i)T^{2} \)
17 \( 1 - 13.5iT - 289T^{2} \)
19 \( 1 + (3.45 - 8.34i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-16.8 + 16.8i)T - 529iT^{2} \)
29 \( 1 + (13.8 - 33.4i)T + (-594. - 594. i)T^{2} \)
31 \( 1 + 24.5iT - 961T^{2} \)
37 \( 1 + (9.89 - 4.09i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-14.4 - 14.4i)T + 1.68e3iT^{2} \)
43 \( 1 + (-17.8 + 7.39i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 43.6T + 2.20e3T^{2} \)
53 \( 1 + (28.0 + 67.7i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-1.70 - 4.10i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (3.53 - 8.53i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (0.300 + 0.124i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-29.0 - 29.0i)T + 5.04e3iT^{2} \)
73 \( 1 + (68.2 + 68.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 67.7T + 6.24e3T^{2} \)
83 \( 1 + (16.4 - 39.5i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (45.3 - 45.3i)T - 7.92e3iT^{2} \)
97 \( 1 + 119.T + 9.40e3T^{2} \)
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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

−11.63773870161684895912891958153, −10.98891284544103966388692233126, −9.636439965507892637111850271721, −8.849424700109607350918503414297, −8.348834518514147611558309793237, −6.70035616562358054873132739243, −5.91319895388537694069012278426, −4.01162614117985234789344821198, −3.44909087924411519077975640815, −1.51965371768348104229840162681, 1.46742282530753521092819354537, 2.97620305615161887918459084928, 4.16833854295263836728980174454, 5.77780509713130903496440580602, 6.87649543561066523793302440914, 7.915566553713142917987227532011, 8.902281806330403210136928511110, 9.527041502436159662382319921624, 10.92977841855741013063690432075, 11.72505650353756976823588080575

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