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Properties

Label 2-2e8-32.21-c3-0-0
Degree $2$
Conductor $256$
Sign $-0.920 - 0.390i$
Analytic cond. $15.1044$
Root an. cond. $3.88644$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.143 − 0.0595i)3-s + (0.767 − 1.85i)5-s + (5.47 + 5.47i)7-s + (−19.0 + 19.0i)9-s + (−36.9 − 15.2i)11-s + (−4.49 − 10.8i)13-s − 0.312i·15-s + 53.8i·17-s + (−31.9 − 77.2i)19-s + (1.11 + 0.461i)21-s + (−50.0 + 50.0i)23-s + (85.5 + 85.5i)25-s + (−3.21 + 7.75i)27-s + (−156. + 64.6i)29-s − 207.·31-s + ⋯
L(s)  = 1  + (0.0276 − 0.0114i)3-s + (0.0686 − 0.165i)5-s + (0.295 + 0.295i)7-s + (−0.706 + 0.706i)9-s + (−1.01 − 0.419i)11-s + (−0.0959 − 0.231i)13-s − 0.00537i·15-s + 0.768i·17-s + (−0.386 − 0.932i)19-s + (0.0115 + 0.00479i)21-s + (−0.454 + 0.454i)23-s + (0.684 + 0.684i)25-s + (−0.0229 + 0.0553i)27-s + (−0.999 + 0.414i)29-s − 1.20·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.920 - 0.390i$
Analytic conductor: \(15.1044\)
Root analytic conductor: \(3.88644\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3/2),\ -0.920 - 0.390i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3975435136\)
\(L(\frac12)\) \(\approx\) \(0.3975435136\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.143 + 0.0595i)T + (19.0 - 19.0i)T^{2} \)
5 \( 1 + (-0.767 + 1.85i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (-5.47 - 5.47i)T + 343iT^{2} \)
11 \( 1 + (36.9 + 15.2i)T + (941. + 941. i)T^{2} \)
13 \( 1 + (4.49 + 10.8i)T + (-1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 - 53.8iT - 4.91e3T^{2} \)
19 \( 1 + (31.9 + 77.2i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (50.0 - 50.0i)T - 1.21e4iT^{2} \)
29 \( 1 + (156. - 64.6i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + 207.T + 2.97e4T^{2} \)
37 \( 1 + (73.7 - 177. i)T + (-3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (293. - 293. i)T - 6.89e4iT^{2} \)
43 \( 1 + (342. + 141. i)T + (5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + 510. iT - 1.03e5T^{2} \)
53 \( 1 + (-590. - 244. i)T + (1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (257. - 622. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (424. - 175. i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (-482. + 199. i)T + (2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (-199. - 199. i)T + 3.57e5iT^{2} \)
73 \( 1 + (127. - 127. i)T - 3.89e5iT^{2} \)
79 \( 1 + 237. iT - 4.93e5T^{2} \)
83 \( 1 + (-9.62 - 23.2i)T + (-4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-329. - 329. i)T + 7.04e5iT^{2} \)
97 \( 1 - 776.T + 9.12e5T^{2} \)
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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

−11.87012201996936527635345506068, −10.99626569335387825561991694493, −10.27699795342153218441857068463, −8.851360219188323367256154465444, −8.256850452262080404157257852129, −7.14981011043784412735715424567, −5.64484115383893292112067381618, −5.03345671069944368562937032175, −3.29857418375282032929846647297, −1.97874546686916627829187074048, 0.14300332020483325274837881788, 2.17063588805245560681774999413, 3.56906343480694755320829426040, 4.94191780403335418803415498878, 6.05518195695264370858701193460, 7.22899559101943106351954150630, 8.190972993941249163366196118051, 9.252305424501758201753477977445, 10.26265454269848320230994728194, 11.11094630419728136281802399158

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