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Properties

Label 2-2e8-16.13-c1-0-1
Degree $2$
Conductor $256$
Sign $-0.130 - 0.991i$
Analytic cond. $2.04417$
Root an. cond. $1.42974$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.517 + 0.517i)3-s + (−1.73 + 1.73i)5-s + 3.86i·7-s − 2.46i·9-s + (−3.34 + 3.34i)11-s + (−0.267 − 0.267i)13-s − 1.79·15-s + 3.46·17-s + (3.34 + 3.34i)19-s + (−1.99 + 1.99i)21-s − 1.79i·23-s − 0.999i·25-s + (2.82 − 2.82i)27-s + (−1.73 − 1.73i)29-s + 5.65·31-s + ⋯
L(s)  = 1  + (0.298 + 0.298i)3-s + (−0.774 + 0.774i)5-s + 1.46i·7-s − 0.821i·9-s + (−1.00 + 1.00i)11-s + (−0.0743 − 0.0743i)13-s − 0.462·15-s + 0.840·17-s + (0.767 + 0.767i)19-s + (−0.436 + 0.436i)21-s − 0.373i·23-s − 0.199i·25-s + (0.544 − 0.544i)27-s + (−0.321 − 0.321i)29-s + 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.130 - 0.991i$
Analytic conductor: \(2.04417\)
Root analytic conductor: \(1.42974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1/2),\ -0.130 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.719357 + 0.820270i\)
\(L(\frac12)\) \(\approx\) \(0.719357 + 0.820270i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.517 - 0.517i)T + 3iT^{2} \)
5 \( 1 + (1.73 - 1.73i)T - 5iT^{2} \)
7 \( 1 - 3.86iT - 7T^{2} \)
11 \( 1 + (3.34 - 3.34i)T - 11iT^{2} \)
13 \( 1 + (0.267 + 0.267i)T + 13iT^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + (-3.34 - 3.34i)T + 19iT^{2} \)
23 \( 1 + 1.79iT - 23T^{2} \)
29 \( 1 + (1.73 + 1.73i)T + 29iT^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + (-3.73 + 3.73i)T - 37iT^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 + (-1.55 + 1.55i)T - 43iT^{2} \)
47 \( 1 + 9.79T + 47T^{2} \)
53 \( 1 + (-7.73 + 7.73i)T - 53iT^{2} \)
59 \( 1 + (-5.13 + 5.13i)T - 59iT^{2} \)
61 \( 1 + (-3.73 - 3.73i)T + 61iT^{2} \)
67 \( 1 + (4.38 + 4.38i)T + 67iT^{2} \)
71 \( 1 + 1.79iT - 71T^{2} \)
73 \( 1 + 2.53iT - 73T^{2} \)
79 \( 1 - 4.14T + 79T^{2} \)
83 \( 1 + (1.55 + 1.55i)T + 83iT^{2} \)
89 \( 1 - 2.53iT - 89T^{2} \)
97 \( 1 - 3.46T + 97T^{2} \)
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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

−12.09892949064119134638216915428, −11.57208722498000959299427546442, −10.17455945360786644492680878133, −9.543110652147263525540073853556, −8.308651941432500904770974424016, −7.51553901784430358792636653960, −6.23676392288393924250976650697, −5.07243271396618011248734780876, −3.55327633893016097727842944189, −2.55318684750207342388014435224, 0.872220906877429281569831727505, 3.08480345280821336825456132144, 4.39323780303103347087637906026, 5.41175964300818403089669839294, 7.17952364823363474904132025884, 7.84498003440506320674270242473, 8.514020312902734723553523393016, 9.983649891249550556373299232876, 10.84277077225791008850352764644, 11.69666739500519817487905597168

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