Properties

Label 2-2e8-256.101-c1-0-3
Degree $2$
Conductor $256$
Sign $0.978 + 0.207i$
Analytic cond. $2.04417$
Root an. cond. $1.42974$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.502 − 1.32i)2-s + (−2.92 + 0.733i)3-s + (−1.49 + 1.32i)4-s + (−3.98 − 1.88i)5-s + (2.44 + 3.50i)6-s + (0.386 + 1.27i)7-s + (2.50 + 1.30i)8-s + (5.38 − 2.87i)9-s + (−0.488 + 6.21i)10-s + (1.23 − 0.183i)11-s + (3.40 − 4.98i)12-s + (0.670 − 1.87i)13-s + (1.49 − 1.15i)14-s + (13.0 + 2.59i)15-s + (0.467 − 3.97i)16-s + (−5.74 + 1.14i)17-s + ⋯
L(s)  = 1  + (−0.355 − 0.934i)2-s + (−1.69 + 0.423i)3-s + (−0.747 + 0.664i)4-s + (−1.78 − 0.843i)5-s + (0.996 + 1.42i)6-s + (0.146 + 0.481i)7-s + (0.886 + 0.462i)8-s + (1.79 − 0.959i)9-s + (−0.154 + 1.96i)10-s + (0.372 − 0.0552i)11-s + (0.981 − 1.43i)12-s + (0.186 − 0.520i)13-s + (0.398 − 0.307i)14-s + (3.36 + 0.670i)15-s + (0.116 − 0.993i)16-s + (−1.39 + 0.277i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.978 + 0.207i$
Analytic conductor: \(2.04417\)
Root analytic conductor: \(1.42974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1/2),\ 0.978 + 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.324578 - 0.0341251i\)
\(L(\frac12)\) \(\approx\) \(0.324578 - 0.0341251i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.502 + 1.32i)T \)
good3 \( 1 + (2.92 - 0.733i)T + (2.64 - 1.41i)T^{2} \)
5 \( 1 + (3.98 + 1.88i)T + (3.17 + 3.86i)T^{2} \)
7 \( 1 + (-0.386 - 1.27i)T + (-5.82 + 3.88i)T^{2} \)
11 \( 1 + (-1.23 + 0.183i)T + (10.5 - 3.19i)T^{2} \)
13 \( 1 + (-0.670 + 1.87i)T + (-10.0 - 8.24i)T^{2} \)
17 \( 1 + (5.74 - 1.14i)T + (15.7 - 6.50i)T^{2} \)
19 \( 1 + (-3.92 - 3.55i)T + (1.86 + 18.9i)T^{2} \)
23 \( 1 + (-3.09 + 0.304i)T + (22.5 - 4.48i)T^{2} \)
29 \( 1 + (0.0217 + 0.0293i)T + (-8.41 + 27.7i)T^{2} \)
31 \( 1 + (2.24 + 0.929i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (0.218 - 4.45i)T + (-36.8 - 3.62i)T^{2} \)
41 \( 1 + (-4.65 + 5.67i)T + (-7.99 - 40.2i)T^{2} \)
43 \( 1 + (1.17 - 4.69i)T + (-37.9 - 20.2i)T^{2} \)
47 \( 1 + (-1.28 - 0.856i)T + (17.9 + 43.4i)T^{2} \)
53 \( 1 + (-3.63 + 4.90i)T + (-15.3 - 50.7i)T^{2} \)
59 \( 1 + (-4.18 - 11.6i)T + (-45.6 + 37.4i)T^{2} \)
61 \( 1 + (-2.67 + 4.45i)T + (-28.7 - 53.7i)T^{2} \)
67 \( 1 + (-1.45 - 0.870i)T + (31.5 + 59.0i)T^{2} \)
71 \( 1 + (1.88 - 3.52i)T + (-39.4 - 59.0i)T^{2} \)
73 \( 1 + (0.486 - 1.60i)T + (-60.6 - 40.5i)T^{2} \)
79 \( 1 + (-1.42 - 2.13i)T + (-30.2 + 72.9i)T^{2} \)
83 \( 1 + (-0.200 - 4.07i)T + (-82.6 + 8.13i)T^{2} \)
89 \( 1 + (-8.15 - 0.803i)T + (87.2 + 17.3i)T^{2} \)
97 \( 1 + (0.182 - 0.439i)T + (-68.5 - 68.5i)T^{2} \)
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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

−11.72325351946711074172496782308, −11.32359898500608663200640609625, −10.51047359053355565759347715573, −9.198897802815783787338643257561, −8.314918165329992879474305039019, −7.15033313929282320622407975321, −5.44940462924575193625835184354, −4.54650691169278583602310142467, −3.72454747637438642121582098920, −0.868674389538496007514038699767, 0.56101402023385195556800829891, 4.09149181773510771825361600912, 4.91271070687932396030096008701, 6.41894164433554480529364423982, 7.07730122207619073445299929064, 7.52244406540743779143665672684, 8.965069355870017896862248299535, 10.54563366684593380945452262269, 11.18674553454281812003746868117, 11.66538970813260492316666367753

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