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Properties

Label 2-2e8-32.29-c3-0-12
Degree $2$
Conductor $256$
Sign $-0.448 + 0.893i$
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  + (−1.65 − 0.686i)3-s + (4.13 + 9.99i)5-s + (−24.2 + 24.2i)7-s + (−16.8 − 16.8i)9-s + (3.73 − 1.54i)11-s + (23.2 − 56.0i)13-s − 19.4i·15-s − 26.9i·17-s + (22.7 − 54.8i)19-s + (56.7 − 23.5i)21-s + (−76.0 − 76.0i)23-s + (5.64 − 5.64i)25-s + (34.8 + 84.1i)27-s + (108. + 44.7i)29-s − 175.·31-s + ⋯
L(s)  = 1  + (−0.318 − 0.132i)3-s + (0.370 + 0.893i)5-s + (−1.30 + 1.30i)7-s + (−0.622 − 0.622i)9-s + (0.102 − 0.0424i)11-s + (0.495 − 1.19i)13-s − 0.333i·15-s − 0.385i·17-s + (0.274 − 0.662i)19-s + (0.589 − 0.244i)21-s + (−0.689 − 0.689i)23-s + (0.0451 − 0.0451i)25-s + (0.248 + 0.599i)27-s + (0.692 + 0.286i)29-s − 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.448 + 0.893i$
Analytic conductor: \(15.1044\)
Root analytic conductor: \(3.88644\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3/2),\ -0.448 + 0.893i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4622090309\)
\(L(\frac12)\) \(\approx\) \(0.4622090309\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.65 + 0.686i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (-4.13 - 9.99i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (24.2 - 24.2i)T - 343iT^{2} \)
11 \( 1 + (-3.73 + 1.54i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (-23.2 + 56.0i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 26.9iT - 4.91e3T^{2} \)
19 \( 1 + (-22.7 + 54.8i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (76.0 + 76.0i)T + 1.21e4iT^{2} \)
29 \( 1 + (-108. - 44.7i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 175.T + 2.97e4T^{2} \)
37 \( 1 + (129. + 311. i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-70.4 - 70.4i)T + 6.89e4iT^{2} \)
43 \( 1 + (103. - 42.7i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 249. iT - 1.03e5T^{2} \)
53 \( 1 + (597. - 247. i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (75.6 + 182. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (-309. - 128. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (297. + 123. i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (675. - 675. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-350. - 350. i)T + 3.89e5iT^{2} \)
79 \( 1 + 564. iT - 4.93e5T^{2} \)
83 \( 1 + (-105. + 254. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (448. - 448. i)T - 7.04e5iT^{2} \)
97 \( 1 + 1.76e3T + 9.12e5T^{2} \)
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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

−11.27301046063028143669801063046, −10.32139281579986869167413185825, −9.355954173697951443979436571880, −8.545015267984979794431111916470, −6.93834446466809723891948255672, −6.14631242738520255322103406854, −5.51587672394018567300440052156, −3.31188521590475570315145062874, −2.63654860044342301853321893963, −0.18894675182366520403235059660, 1.45927639679580160431881870272, 3.49062441158284781963503187177, 4.57101900175250695755274144768, 5.86041949885358757895598705008, 6.75127028035250179859758150358, 8.007829200145585284567378720962, 9.173993072878046427014018931600, 9.945568329518830319058622473760, 10.83767417949542817487646441387, 11.90215185849290077806241421649

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