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Properties

Label 2-2e8-16.11-c2-0-12
Degree $2$
Conductor $256$
Sign $0.382 + 0.923i$
Analytic cond. $6.97549$
Root an. cond. $2.64111$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.37 + 2.37i)3-s + (−6.35 − 6.35i)5-s − 1.92·7-s + 2.28i·9-s + (10.8 − 10.8i)11-s + (8.35 − 8.35i)13-s − 30.2i·15-s − 0.717·17-s + (−21.2 − 21.2i)19-s + (−4.56 − 4.56i)21-s + 22.7·23-s + 55.8i·25-s + (15.9 − 15.9i)27-s + (−7.79 + 7.79i)29-s + 13.3i·31-s + ⋯
L(s)  = 1  + (0.791 + 0.791i)3-s + (−1.27 − 1.27i)5-s − 0.274·7-s + 0.253i·9-s + (0.987 − 0.987i)11-s + (0.642 − 0.642i)13-s − 2.01i·15-s − 0.0422·17-s + (−1.11 − 1.11i)19-s + (−0.217 − 0.217i)21-s + 0.988·23-s + 2.23i·25-s + (0.590 − 0.590i)27-s + (−0.268 + 0.268i)29-s + 0.430i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(6.97549\)
Root analytic conductor: \(2.64111\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1),\ 0.382 + 0.923i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.22324 - 0.817347i\)
\(L(\frac12)\) \(\approx\) \(1.22324 - 0.817347i\)
\(L(2)\) 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 \( 1 + (-2.37 - 2.37i)T + 9iT^{2} \)
5 \( 1 + (6.35 + 6.35i)T + 25iT^{2} \)
7 \( 1 + 1.92T + 49T^{2} \)
11 \( 1 + (-10.8 + 10.8i)T - 121iT^{2} \)
13 \( 1 + (-8.35 + 8.35i)T - 169iT^{2} \)
17 \( 1 + 0.717T + 289T^{2} \)
19 \( 1 + (21.2 + 21.2i)T + 361iT^{2} \)
23 \( 1 - 22.7T + 529T^{2} \)
29 \( 1 + (7.79 - 7.79i)T - 841iT^{2} \)
31 \( 1 - 13.3iT - 961T^{2} \)
37 \( 1 + (29.0 + 29.0i)T + 1.36e3iT^{2} \)
41 \( 1 + 1.43iT - 1.68e3T^{2} \)
43 \( 1 + (20.2 - 20.2i)T - 1.84e3iT^{2} \)
47 \( 1 + 16.9iT - 2.20e3T^{2} \)
53 \( 1 + (-22.9 - 22.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (-47.8 + 47.8i)T - 3.48e3iT^{2} \)
61 \( 1 + (3.64 - 3.64i)T - 3.72e3iT^{2} \)
67 \( 1 + (-27.8 - 27.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 52.6T + 5.04e3T^{2} \)
73 \( 1 + 65.2iT - 5.32e3T^{2} \)
79 \( 1 - 79.6iT - 6.24e3T^{2} \)
83 \( 1 + (-26.8 - 26.8i)T + 6.88e3iT^{2} \)
89 \( 1 - 23.0iT - 7.92e3T^{2} \)
97 \( 1 - 47.2T + 9.40e3T^{2} \)
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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

−11.55486312300936600381420440792, −10.75933342712390164742658547388, −9.220862606816937661378863668005, −8.797406468630487621715155118485, −8.171819626404617736671029354096, −6.67323715862939796533496415620, −5.07612210418992446582418004226, −3.99193734575602345728898081557, −3.32780018003076378445299626940, −0.72879574303436158189139487169, 1.91920005714527517779141845928, 3.31695185744563903338097980491, 4.22631328033281865922131390178, 6.53663524577910130251374560538, 7.00150494249266154350073767049, 7.939779869759258787665955021328, 8.794887891218590580184654284109, 10.14127532737926589260044586435, 11.16702716565231526179123833236, 11.97218359009499373517162601575

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