Properties

Label 2-2e8-4.3-c4-0-2
Degree $2$
Conductor $256$
Sign $-1$
Analytic cond. $26.4627$
Root an. cond. $5.14419$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.79i·3-s + 8·5-s − 78.3i·7-s − 14.9·9-s + 107. i·11-s − 216·13-s + 78.3i·15-s − 162·17-s + 440. i·19-s + 767.·21-s + 705. i·23-s − 561·25-s + 646. i·27-s − 1.30e3·29-s − 627. i·31-s + ⋯
L(s)  = 1  + 1.08i·3-s + 0.320·5-s − 1.59i·7-s − 0.185·9-s + 0.890i·11-s − 1.27·13-s + 0.348i·15-s − 0.560·17-s + 1.22i·19-s + 1.74·21-s + 1.33i·23-s − 0.897·25-s + 0.887i·27-s − 1.55·29-s − 0.652i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-1$
Analytic conductor: \(26.4627\)
Root analytic conductor: \(5.14419\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :2),\ -1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.6513819986\)
\(L(\frac12)\) \(\approx\) \(0.6513819986\)
\(L(3)\) 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 \( 1 - 9.79iT - 81T^{2} \)
5 \( 1 - 8T + 625T^{2} \)
7 \( 1 + 78.3iT - 2.40e3T^{2} \)
11 \( 1 - 107. iT - 1.46e4T^{2} \)
13 \( 1 + 216T + 2.85e4T^{2} \)
17 \( 1 + 162T + 8.35e4T^{2} \)
19 \( 1 - 440. iT - 1.30e5T^{2} \)
23 \( 1 - 705. iT - 2.79e5T^{2} \)
29 \( 1 + 1.30e3T + 7.07e5T^{2} \)
31 \( 1 + 627. iT - 9.23e5T^{2} \)
37 \( 1 - 1.51e3T + 1.87e6T^{2} \)
41 \( 1 + 1.89e3T + 2.82e6T^{2} \)
43 \( 1 + 2.90e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.41e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.97e3T + 7.89e6T^{2} \)
59 \( 1 - 2.26e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.37e3T + 1.38e7T^{2} \)
67 \( 1 + 1.67e3iT - 2.01e7T^{2} \)
71 \( 1 - 7.75e3iT - 2.54e7T^{2} \)
73 \( 1 + 2.75e3T + 2.83e7T^{2} \)
79 \( 1 + 7.99e3iT - 3.89e7T^{2} \)
83 \( 1 - 9.33e3iT - 4.74e7T^{2} \)
89 \( 1 + 2.43e3T + 6.27e7T^{2} \)
97 \( 1 - 7.45e3T + 8.85e7T^{2} \)
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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

−11.63025970979512554636336158002, −10.59364013862503193177692896789, −9.826357163803286981870802320120, −9.550558628259201686936193955710, −7.70593043828170453803768846155, −7.11451864704584627079397822352, −5.49516412405161868837684853169, −4.36812397703603333036581892109, −3.73381101785864124522119529141, −1.78935071864271190590490159475, 0.19434025542422163061740361424, 1.98076425987799346323449231400, 2.74813191536377158874766520878, 4.86147807191654725722724494305, 5.96220424606043269132645582252, 6.77455746569168342120232063002, 7.940540475249908386288108114826, 8.866795081254952933762927723174, 9.715269151571272580927005741142, 11.20969892195562313090055100587

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