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Properties

Label 2-2e8-32.3-c2-0-5
Degree $2$
Conductor $256$
Sign $0.951 - 0.306i$
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  + (0.527 − 1.27i)3-s + (0.642 + 1.55i)5-s + (4.95 + 4.95i)7-s + (5.01 + 5.01i)9-s + (−4.27 − 10.3i)11-s + (−1.68 + 4.06i)13-s + 2.31·15-s + 28.6i·17-s + (17.5 + 7.26i)19-s + (8.91 − 3.69i)21-s + (24.3 − 24.3i)23-s + (15.6 − 15.6i)25-s + (20.5 − 8.49i)27-s + (−8.57 − 3.55i)29-s − 5.73i·31-s + ⋯
L(s)  = 1  + (0.175 − 0.424i)3-s + (0.128 + 0.310i)5-s + (0.707 + 0.707i)7-s + (0.557 + 0.557i)9-s + (−0.388 − 0.937i)11-s + (−0.129 + 0.312i)13-s + 0.154·15-s + 1.68i·17-s + (0.923 + 0.382i)19-s + (0.424 − 0.175i)21-s + (1.05 − 1.05i)23-s + (0.627 − 0.627i)25-s + (0.759 − 0.314i)27-s + (−0.295 − 0.122i)29-s − 0.184i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.951 - 0.306i$
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: \((2,\ 256,\ (\ :1),\ 0.951 - 0.306i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.87204 + 0.294123i\)
\(L(\frac12)\) \(\approx\) \(1.87204 + 0.294123i\)
\(L(2)\) 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 \( 1 + (-0.527 + 1.27i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (-0.642 - 1.55i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (-4.95 - 4.95i)T + 49iT^{2} \)
11 \( 1 + (4.27 + 10.3i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (1.68 - 4.06i)T + (-119. - 119. i)T^{2} \)
17 \( 1 - 28.6iT - 289T^{2} \)
19 \( 1 + (-17.5 - 7.26i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (-24.3 + 24.3i)T - 529iT^{2} \)
29 \( 1 + (8.57 + 3.55i)T + (594. + 594. i)T^{2} \)
31 \( 1 + 5.73iT - 961T^{2} \)
37 \( 1 + (-26.1 - 63.0i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (14.2 + 14.2i)T + 1.68e3iT^{2} \)
43 \( 1 + (10.1 + 24.4i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 57.9T + 2.20e3T^{2} \)
53 \( 1 + (-46.3 + 19.2i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (27.6 - 11.4i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (76.3 + 31.6i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (36.1 - 87.3i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-5.39 - 5.39i)T + 5.04e3iT^{2} \)
73 \( 1 + (25.4 + 25.4i)T + 5.32e3iT^{2} \)
79 \( 1 + 50.1T + 6.24e3T^{2} \)
83 \( 1 + (100. + 41.7i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-10.6 + 10.6i)T - 7.92e3iT^{2} \)
97 \( 1 + 14.3T + 9.40e3T^{2} \)
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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

−11.84818084886799452492115622558, −10.87274462140414461555522761966, −10.10633618178364526112508553237, −8.580986589055913350825659988684, −8.138100620791382939844701941349, −6.87173028552651093303036123831, −5.78407939981207416075660899517, −4.61559368943880470134897152598, −2.92148925009704427103482174014, −1.58816949405973478930991610989, 1.17291646197327495313710594098, 3.11068294818569842879581958922, 4.57295585667737169568096107316, 5.21459052930118710275455941759, 7.14003488731220273650262098756, 7.54314189830579847004403123377, 9.210265018745754779322009445274, 9.622721362939860092756451475170, 10.77665102240238916795358552233, 11.65233834271305056358583410704

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