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Properties

Label 2-2e8-4.3-c10-0-34
Degree $2$
Conductor $256$
Sign $-i$
Analytic cond. $162.651$
Root an. cond. $12.7534$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 104. i·3-s + 4.60e3·5-s + 2.39e4i·7-s + 4.81e4·9-s − 2.16e5i·11-s − 6.41e5·13-s + 4.81e5i·15-s + 9.24e5·17-s − 2.63e6i·19-s − 2.50e6·21-s + 2.37e6i·23-s + 1.14e7·25-s + 1.12e7i·27-s − 6.16e6·29-s + 2.62e7i·31-s + ⋯
L(s)  = 1  + 0.430i·3-s + 1.47·5-s + 1.42i·7-s + 0.814·9-s − 1.34i·11-s − 1.72·13-s + 0.633i·15-s + 0.650·17-s − 1.06i·19-s − 0.612·21-s + 0.368i·23-s + 1.17·25-s + 0.780i·27-s − 0.300·29-s + 0.917i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-i$
Analytic conductor: \(162.651\)
Root analytic conductor: \(12.7534\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :5),\ -i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(3.039061458\)
\(L(\frac12)\) \(\approx\) \(3.039061458\)
\(L(6)\) 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 \( 1 - 104. iT - 5.90e4T^{2} \)
5 \( 1 - 4.60e3T + 9.76e6T^{2} \)
7 \( 1 - 2.39e4iT - 2.82e8T^{2} \)
11 \( 1 + 2.16e5iT - 2.59e10T^{2} \)
13 \( 1 + 6.41e5T + 1.37e11T^{2} \)
17 \( 1 - 9.24e5T + 2.01e12T^{2} \)
19 \( 1 + 2.63e6iT - 6.13e12T^{2} \)
23 \( 1 - 2.37e6iT - 4.14e13T^{2} \)
29 \( 1 + 6.16e6T + 4.20e14T^{2} \)
31 \( 1 - 2.62e7iT - 8.19e14T^{2} \)
37 \( 1 - 2.49e7T + 4.80e15T^{2} \)
41 \( 1 - 2.23e8T + 1.34e16T^{2} \)
43 \( 1 - 2.18e8iT - 2.16e16T^{2} \)
47 \( 1 - 1.78e8iT - 5.25e16T^{2} \)
53 \( 1 - 1.65e8T + 1.74e17T^{2} \)
59 \( 1 - 2.04e8iT - 5.11e17T^{2} \)
61 \( 1 + 1.00e9T + 7.13e17T^{2} \)
67 \( 1 - 1.16e9iT - 1.82e18T^{2} \)
71 \( 1 + 1.04e8iT - 3.25e18T^{2} \)
73 \( 1 - 5.20e8T + 4.29e18T^{2} \)
79 \( 1 - 4.06e9iT - 9.46e18T^{2} \)
83 \( 1 + 4.48e9iT - 1.55e19T^{2} \)
89 \( 1 - 4.01e9T + 3.11e19T^{2} \)
97 \( 1 - 7.96e9T + 7.37e19T^{2} \)
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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

−10.24974005625476981919468707834, −9.411844723085047874632955688211, −9.044324906929248781266420893868, −7.57731980224026985717511677789, −6.26193279612475989626049190447, −5.51314963422173076685263051641, −4.76845547056551047907868106228, −2.96940622592326268268743853222, −2.31613375269402895731768070314, −1.08565078129703599436915448476, 0.57545484770198740274822733179, 1.68540646466321649301493428505, 2.29581649444809850265144531763, 4.05622060456957399734473229569, 4.95133009855939172064540429556, 6.17371838162571321975909343547, 7.32577963401030905748603094021, 7.52611787068686294933589356546, 9.548123741622396030298287872967, 9.954489074383268097447191359522

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