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Properties

Label 2-2e8-32.11-c2-0-1
Degree $2$
Conductor $256$
Sign $0.489 - 0.871i$
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  + (−2.10 − 5.07i)3-s + (−1.74 + 4.21i)5-s + (−0.392 + 0.392i)7-s + (−14.9 + 14.9i)9-s + (−2.90 + 7.02i)11-s + (4.50 + 10.8i)13-s + 25.0·15-s − 10.5i·17-s + (1.88 − 0.781i)19-s + (2.81 + 1.16i)21-s + (−0.445 − 0.445i)23-s + (2.94 + 2.94i)25-s + (61.7 + 25.5i)27-s + (−0.741 + 0.307i)29-s + 47.6i·31-s + ⋯
L(s)  = 1  + (−0.700 − 1.69i)3-s + (−0.349 + 0.843i)5-s + (−0.0560 + 0.0560i)7-s + (−1.66 + 1.66i)9-s + (−0.264 + 0.638i)11-s + (0.346 + 0.836i)13-s + 1.67·15-s − 0.620i·17-s + (0.0993 − 0.0411i)19-s + (0.134 + 0.0555i)21-s + (−0.0193 − 0.0193i)23-s + (0.117 + 0.117i)25-s + (2.28 + 0.947i)27-s + (−0.0255 + 0.0105i)29-s + 1.53i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.489 - 0.871i$
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: \((2,\ 256,\ (\ :1),\ 0.489 - 0.871i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.553574 + 0.324008i\)
\(L(\frac12)\) \(\approx\) \(0.553574 + 0.324008i\)
\(L(2)\) 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 \( 1 + (2.10 + 5.07i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (1.74 - 4.21i)T + (-17.6 - 17.6i)T^{2} \)
7 \( 1 + (0.392 - 0.392i)T - 49iT^{2} \)
11 \( 1 + (2.90 - 7.02i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (-4.50 - 10.8i)T + (-119. + 119. i)T^{2} \)
17 \( 1 + 10.5iT - 289T^{2} \)
19 \( 1 + (-1.88 + 0.781i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (0.445 + 0.445i)T + 529iT^{2} \)
29 \( 1 + (0.741 - 0.307i)T + (594. - 594. i)T^{2} \)
31 \( 1 - 47.6iT - 961T^{2} \)
37 \( 1 + (14.5 - 35.0i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (11.3 - 11.3i)T - 1.68e3iT^{2} \)
43 \( 1 + (-14.6 + 35.3i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 80.5T + 2.20e3T^{2} \)
53 \( 1 + (-66.6 - 27.5i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (65.0 + 26.9i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (87.4 - 36.2i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-7.12 - 17.1i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (14.8 - 14.8i)T - 5.04e3iT^{2} \)
73 \( 1 + (-18.6 + 18.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 36.2T + 6.24e3T^{2} \)
83 \( 1 + (27.0 - 11.2i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (56.4 + 56.4i)T + 7.92e3iT^{2} \)
97 \( 1 - 158.T + 9.40e3T^{2} \)
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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

−11.93501727680328617557211864650, −11.31601429221581724778976902801, −10.37804548877173616592448539809, −8.814141955693205643139782569591, −7.59160419478772591529035287117, −6.98932436607302750332839746491, −6.28547425466746675436972039857, −4.95083535221165223889684771078, −2.93935238138446686532675264723, −1.55611392557381774676077799271, 0.37068654615484126600275830755, 3.37210646949747033196524747262, 4.33357134623293650683276421518, 5.31947023627154192026164136217, 6.09974564191576681226379522580, 8.061231604305203091020729933059, 8.883068002421461858478784937519, 9.845718585922087965467710627675, 10.68669340224311466572336874840, 11.37853056808666270517840081108

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