Properties

Label 2-2e8-8.3-c6-0-31
Degree $2$
Conductor $256$
Sign $0.707 - 0.707i$
Analytic cond. $58.8938$
Root an. cond. $7.67423$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 51.1·3-s + 142. i·5-s − 217. i·7-s + 1.89e3·9-s − 25.7·11-s + 1.39e3i·13-s + 7.30e3i·15-s + 4.00e3·17-s − 129.·19-s − 1.11e4i·21-s + 3.69e3i·23-s − 4.75e3·25-s + 5.95e4·27-s − 4.08e4i·29-s + 4.61e4i·31-s + ⋯
L(s)  = 1  + 1.89·3-s + 1.14i·5-s − 0.634i·7-s + 2.59·9-s − 0.0193·11-s + 0.632i·13-s + 2.16i·15-s + 0.814·17-s − 0.0188·19-s − 1.20i·21-s + 0.303i·23-s − 0.304·25-s + 3.02·27-s − 1.67i·29-s + 1.54i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(58.8938\)
Root analytic conductor: \(7.67423\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(4.701062642\)
\(L(\frac12)\) \(\approx\) \(4.701062642\)
\(L(4)\) 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 \( 1 - 51.1T + 729T^{2} \)
5 \( 1 - 142. iT - 1.56e4T^{2} \)
7 \( 1 + 217. iT - 1.17e5T^{2} \)
11 \( 1 + 25.7T + 1.77e6T^{2} \)
13 \( 1 - 1.39e3iT - 4.82e6T^{2} \)
17 \( 1 - 4.00e3T + 2.41e7T^{2} \)
19 \( 1 + 129.T + 4.70e7T^{2} \)
23 \( 1 - 3.69e3iT - 1.48e8T^{2} \)
29 \( 1 + 4.08e4iT - 5.94e8T^{2} \)
31 \( 1 - 4.61e4iT - 8.87e8T^{2} \)
37 \( 1 - 6.59e4iT - 2.56e9T^{2} \)
41 \( 1 - 4.18e4T + 4.75e9T^{2} \)
43 \( 1 - 3.00e4T + 6.32e9T^{2} \)
47 \( 1 - 7.97e4iT - 1.07e10T^{2} \)
53 \( 1 - 1.37e4iT - 2.21e10T^{2} \)
59 \( 1 + 3.00e5T + 4.21e10T^{2} \)
61 \( 1 + 2.43e5iT - 5.15e10T^{2} \)
67 \( 1 + 4.06e5T + 9.04e10T^{2} \)
71 \( 1 - 2.49e5iT - 1.28e11T^{2} \)
73 \( 1 - 3.66e5T + 1.51e11T^{2} \)
79 \( 1 + 4.52e5iT - 2.43e11T^{2} \)
83 \( 1 + 5.12e5T + 3.26e11T^{2} \)
89 \( 1 - 7.97e5T + 4.96e11T^{2} \)
97 \( 1 + 8.78e3T + 8.32e11T^{2} \)
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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

−10.74368210873826673448095364711, −9.988720665499127036194650614735, −9.188280588568109953567260021533, −8.026577255000933807437431453451, −7.38687941154751441608152317490, −6.50112816979013301337246998974, −4.40924151151587955730634638419, −3.41164469763660828301806103448, −2.66868973021685771431835726803, −1.42679877506258174201580861230, 1.01132917359906854116457908149, 2.19208899178517823929466567952, 3.26171066802670981283651099175, 4.38424293751647697918565588177, 5.59330388505684489026190098384, 7.35803536112275390991379017749, 8.139131114503097551912728951054, 8.928636712332356440518384333677, 9.400168487341648504820316960734, 10.53063895242180312985831179442

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