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Properties

Label 2-2e8-64.21-c1-0-2
Degree $2$
Conductor $256$
Sign $0.322 - 0.946i$
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  + (0.435 + 2.18i)3-s + (0.649 − 0.434i)5-s + (3.64 + 1.50i)7-s + (−1.82 + 0.757i)9-s + (−5.80 − 1.15i)11-s + (2.03 + 1.35i)13-s + (1.23 + 1.23i)15-s + (0.960 − 0.960i)17-s + (−0.435 + 0.652i)19-s + (−1.71 + 8.63i)21-s + (−0.421 − 1.01i)23-s + (−1.67 + 4.05i)25-s + (1.26 + 1.89i)27-s + (−1.43 + 0.286i)29-s − 6.88i·31-s + ⋯
L(s)  = 1  + (0.251 + 1.26i)3-s + (0.290 − 0.194i)5-s + (1.37 + 0.570i)7-s + (−0.609 + 0.252i)9-s + (−1.74 − 0.347i)11-s + (0.564 + 0.377i)13-s + (0.318 + 0.318i)15-s + (0.232 − 0.232i)17-s + (−0.100 + 0.149i)19-s + (−0.374 + 1.88i)21-s + (−0.0879 − 0.212i)23-s + (−0.335 + 0.811i)25-s + (0.243 + 0.364i)27-s + (−0.267 + 0.0531i)29-s − 1.23i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.322 - 0.946i$
Analytic conductor: \(2.04417\)
Root analytic conductor: \(1.42974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1/2),\ 0.322 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22086 + 0.873667i\)
\(L(\frac12)\) \(\approx\) \(1.22086 + 0.873667i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-0.435 - 2.18i)T + (-2.77 + 1.14i)T^{2} \)
5 \( 1 + (-0.649 + 0.434i)T + (1.91 - 4.61i)T^{2} \)
7 \( 1 + (-3.64 - 1.50i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (5.80 + 1.15i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + (-2.03 - 1.35i)T + (4.97 + 12.0i)T^{2} \)
17 \( 1 + (-0.960 + 0.960i)T - 17iT^{2} \)
19 \( 1 + (0.435 - 0.652i)T + (-7.27 - 17.5i)T^{2} \)
23 \( 1 + (0.421 + 1.01i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (1.43 - 0.286i)T + (26.7 - 11.0i)T^{2} \)
31 \( 1 + 6.88iT - 31T^{2} \)
37 \( 1 + (1.07 + 1.61i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (2.79 + 6.74i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-1.20 + 6.04i)T + (-39.7 - 16.4i)T^{2} \)
47 \( 1 + (-6.30 + 6.30i)T - 47iT^{2} \)
53 \( 1 + (-10.2 - 2.04i)T + (48.9 + 20.2i)T^{2} \)
59 \( 1 + (4.21 - 2.81i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (2.08 + 10.4i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (1.42 + 7.14i)T + (-61.8 + 25.6i)T^{2} \)
71 \( 1 + (2.49 + 1.03i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (1.90 - 0.789i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-6.20 - 6.20i)T + 79iT^{2} \)
83 \( 1 + (1.13 - 1.70i)T + (-31.7 - 76.6i)T^{2} \)
89 \( 1 + (-1.15 + 2.79i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 - 13.1iT - 97T^{2} \)
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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

−12.00606076945688036796606426159, −10.95479616146796343928451949913, −10.44293633216596631004843491809, −9.282375630856488010254438489922, −8.513301574694536092620914494009, −7.60198633691239277863193172083, −5.58237722767433077114855307061, −5.08061600217496601242847731758, −3.84017181434496810892394233875, −2.25089938675213312400826079456, 1.42068178687241768572078799980, 2.65912533673665158914248771564, 4.63264406497127080227948413703, 5.80641482800001239627358766630, 7.14155602992267026232951199277, 7.87771077119393605359611202473, 8.410798705559270207062643409839, 10.20884775731575249206022965905, 10.82932381995781099501383223593, 11.96139519729803158642878647277

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