Properties

Degree $2$
Conductor $256$
Sign $0.707 + 0.707i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·3-s + 2i·5-s + 4·7-s − 9-s − 2i·11-s − 2i·13-s + 4·15-s − 2·17-s − 2i·19-s − 8i·21-s − 4·23-s + 25-s − 4i·27-s + 6i·29-s − 4·33-s + ⋯
L(s)  = 1  − 1.15i·3-s + 0.894i·5-s + 1.51·7-s − 0.333·9-s − 0.603i·11-s − 0.554i·13-s + 1.03·15-s − 0.485·17-s − 0.458i·19-s − 1.74i·21-s − 0.834·23-s + 0.200·25-s − 0.769i·27-s + 1.11i·29-s − 0.696·33-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.707 + 0.707i$
Motivic weight: \(1\)
Character: $\chi_{256} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31847 - 0.546129i\)
\(L(\frac12)\) \(\approx\) \(1.31847 - 0.546129i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 3T^{2} \)
5 \( 1 - 2iT - 5T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 14iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 + 10iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−11.75762837471476313294134706289, −11.15648399208933492725911632942, −10.26577727041265197797122910912, −8.611783944663925494202640188699, −7.86528301643183403367527227627, −7.03622164116349230603131877481, −6.05194408029281576944524179606, −4.67760979703143973951450660168, −2.86673688421774418436559376471, −1.48527687479949932962438323565, 1.84807485602538521290021290184, 4.18750880843649192496613275121, 4.59266733615629306970623763094, 5.65174324111818920691717656115, 7.42675807818385756312028625007, 8.468954768123919982906953682469, 9.256108972438302022834978407199, 10.19846153151799484232110380476, 11.14494699170078649725293233429, 11.95229495174916424715383421785

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