Properties

Degree $2$
Conductor $256$
Sign $-0.831 - 0.555i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.292i)3-s + (−1.12 + 2.70i)5-s + (−1 − i)7-s + (−1.70 + 1.70i)9-s + (−4.12 − 1.70i)11-s + (−0.292 − 0.707i)13-s − 2.24i·15-s + 2.82i·17-s + (1.53 + 3.70i)19-s + (1 + 0.414i)21-s + (−5.82 + 5.82i)23-s + (−2.53 − 2.53i)25-s + (1.58 − 3.82i)27-s + (3.12 − 1.29i)29-s + 4·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.169i)3-s + (−0.501 + 1.21i)5-s + (−0.377 − 0.377i)7-s + (−0.569 + 0.569i)9-s + (−1.24 − 0.514i)11-s + (−0.0812 − 0.196i)13-s − 0.579i·15-s + 0.685i·17-s + (0.352 + 0.850i)19-s + (0.218 + 0.0903i)21-s + (−1.21 + 1.21i)23-s + (−0.507 − 0.507i)25-s + (0.305 − 0.736i)27-s + (0.579 − 0.240i)29-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.831 - 0.555i$
Motivic weight: \(1\)
Character: $\chi_{256} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1/2),\ -0.831 - 0.555i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148480 + 0.489472i\)
\(L(\frac12)\) \(\approx\) \(0.148480 + 0.489472i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.292i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (1.12 - 2.70i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + (4.12 + 1.70i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.292 + 0.707i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 + (-1.53 - 3.70i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (5.82 - 5.82i)T - 23iT^{2} \)
29 \( 1 + (-3.12 + 1.29i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (0.292 - 0.707i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.171 - 0.171i)T - 41iT^{2} \)
43 \( 1 + (-4.70 - 1.94i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 0.343iT - 47T^{2} \)
53 \( 1 + (-1.12 - 0.464i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (1.87 - 4.53i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (1.70 - 0.707i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (5.53 - 2.29i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-5.82 - 5.82i)T + 71iT^{2} \)
73 \( 1 + (-7 + 7i)T - 73iT^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + (-1.87 - 4.53i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-8.65 - 8.65i)T + 89iT^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−12.18811891723838977520727813663, −11.27197003809940443775368207116, −10.53498331678927994968353536776, −9.998490241513436202873929414806, −8.171849489025051343610788359363, −7.63025063119180227594362763050, −6.30909732504273133543872004360, −5.42310374656439398486846531162, −3.79262279164531121409164249491, −2.69938179409490238225491143709, 0.40080424651910957333917738776, 2.71823636730398433863631993334, 4.49884857309430869895800454415, 5.33984877482147379523385941382, 6.52519675176320441773628871847, 7.81784740065322504792171649381, 8.724129027364337090362334196950, 9.574974066438473647146197724698, 10.78082901688014066988479638249, 12.04171276902232773098667983017

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