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Properties

Label 2-2e8-16.5-c5-0-22
Degree $2$
Conductor $256$
Sign $0.793 + 0.608i$
Analytic cond. $41.0582$
Root an. cond. $6.40767$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (17.5 − 17.5i)3-s + (−43.4 − 43.4i)5-s + 227. i·7-s − 374. i·9-s + (483. + 483. i)11-s + (177. − 177. i)13-s − 1.52e3·15-s + 575.·17-s + (1.39e3 − 1.39e3i)19-s + (4.00e3 + 4.00e3i)21-s + 2.67e3i·23-s + 657. i·25-s + (−2.30e3 − 2.30e3i)27-s + (2.91e3 − 2.91e3i)29-s + 2.71e3·31-s + ⋯
L(s)  = 1  + (1.12 − 1.12i)3-s + (−0.777 − 0.777i)5-s + 1.75i·7-s − 1.54i·9-s + (1.20 + 1.20i)11-s + (0.290 − 0.290i)13-s − 1.75·15-s + 0.482·17-s + (0.889 − 0.889i)19-s + (1.98 + 1.98i)21-s + 1.05i·23-s + 0.210i·25-s + (−0.609 − 0.609i)27-s + (0.642 − 0.642i)29-s + 0.507·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(41.0582\)
Root analytic conductor: \(6.40767\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :5/2),\ 0.793 + 0.608i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.018053954\)
\(L(\frac12)\) \(\approx\) \(3.018053954\)
\(L(\frac{7}{2})\) 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 \( 1 + (-17.5 + 17.5i)T - 243iT^{2} \)
5 \( 1 + (43.4 + 43.4i)T + 3.12e3iT^{2} \)
7 \( 1 - 227. iT - 1.68e4T^{2} \)
11 \( 1 + (-483. - 483. i)T + 1.61e5iT^{2} \)
13 \( 1 + (-177. + 177. i)T - 3.71e5iT^{2} \)
17 \( 1 - 575.T + 1.41e6T^{2} \)
19 \( 1 + (-1.39e3 + 1.39e3i)T - 2.47e6iT^{2} \)
23 \( 1 - 2.67e3iT - 6.43e6T^{2} \)
29 \( 1 + (-2.91e3 + 2.91e3i)T - 2.05e7iT^{2} \)
31 \( 1 - 2.71e3T + 2.86e7T^{2} \)
37 \( 1 + (1.42e3 + 1.42e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 4.36e3iT - 1.15e8T^{2} \)
43 \( 1 + (-9.39e3 - 9.39e3i)T + 1.47e8iT^{2} \)
47 \( 1 + 2.55e3T + 2.29e8T^{2} \)
53 \( 1 + (5.32e3 + 5.32e3i)T + 4.18e8iT^{2} \)
59 \( 1 + (1.41e4 + 1.41e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (6.91e3 - 6.91e3i)T - 8.44e8iT^{2} \)
67 \( 1 + (3.78e4 - 3.78e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 2.02e4iT - 1.80e9T^{2} \)
73 \( 1 + 6.54e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.01e5T + 3.07e9T^{2} \)
83 \( 1 + (-6.65e4 + 6.65e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 7.06e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.02e5T + 8.58e9T^{2} \)
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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

−11.74755048615270616652996510947, −9.473714412744864616947956278492, −9.025404584595445555656425777432, −8.150184428531746570972499819762, −7.38802121857277091362190003730, −6.19064869348299609315313715994, −4.79350216866068163461616510412, −3.30355333050523665021225146311, −2.15381606716216261204832206688, −1.07362783054576423403283203204, 0.985844414479985934905650786580, 3.28383969197038173618489546079, 3.61364186500947137438140790171, 4.47879704137842014856362632212, 6.45647183742741068434354990897, 7.53077610465146437188442405333, 8.349401133772086492236005435900, 9.384323061834815428638032330786, 10.41888536694062244448212753103, 10.86333460655117530744809026068

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