Properties

Label 2-2e8-16.13-c1-0-7
Degree $2$
Conductor $256$
Sign $-0.793 + 0.608i$
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  + (−1.93 − 1.93i)3-s + (1.73 − 1.73i)5-s − 1.03i·7-s + 4.46i·9-s + (−0.896 + 0.896i)11-s + (−3.73 − 3.73i)13-s − 6.69·15-s − 3.46·17-s + (0.896 + 0.896i)19-s + (−1.99 + 1.99i)21-s − 6.69i·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  + (−1.11 − 1.11i)3-s + (0.774 − 0.774i)5-s − 0.391i·7-s + 1.48i·9-s + (−0.270 + 0.270i)11-s + (−1.03 − 1.03i)13-s − 1.72·15-s − 0.840·17-s + (0.205 + 0.205i)19-s + (−0.436 + 0.436i)21-s − 1.39i·23-s − 0.199i·25-s + (0.544 − 0.544i)27-s + (0.321 + 0.321i)29-s + 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.793 + 0.608i$
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: \((2,\ 256,\ (\ :1/2),\ -0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.255354 - 0.752250i\)
\(L(\frac12)\) \(\approx\) \(0.255354 - 0.752250i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.93 + 1.93i)T + 3iT^{2} \)
5 \( 1 + (-1.73 + 1.73i)T - 5iT^{2} \)
7 \( 1 + 1.03iT - 7T^{2} \)
11 \( 1 + (0.896 - 0.896i)T - 11iT^{2} \)
13 \( 1 + (3.73 + 3.73i)T + 13iT^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + (-0.896 - 0.896i)T + 19iT^{2} \)
23 \( 1 + 6.69iT - 23T^{2} \)
29 \( 1 + (-1.73 - 1.73i)T + 29iT^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + (-0.267 + 0.267i)T - 37iT^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + (5.79 - 5.79i)T - 43iT^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 + (-4.26 + 4.26i)T - 53iT^{2} \)
59 \( 1 + (-7.58 + 7.58i)T - 59iT^{2} \)
61 \( 1 + (-0.267 - 0.267i)T + 61iT^{2} \)
67 \( 1 + (-2.96 - 2.96i)T + 67iT^{2} \)
71 \( 1 + 6.69iT - 71T^{2} \)
73 \( 1 + 9.46iT - 73T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 + (-5.79 - 5.79i)T + 83iT^{2} \)
89 \( 1 - 9.46iT - 89T^{2} \)
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

−11.92410900943077530670788886137, −10.70886682922441229102788464354, −9.968412493235729819650876280254, −8.594676759688423078416961369091, −7.46665484037801995641063392856, −6.57592325111436823813393850702, −5.53652848204640845723539778254, −4.75394476549934318326248513629, −2.27455952365988890164315916108, −0.70366005938028500167467444847, 2.52861159596849900746909822271, 4.23496479043453044143770930483, 5.28932573188789348279150091371, 6.15228240612533564927332881547, 7.12540700073964673204402846820, 8.930516941524796120170125468570, 9.861670130422958862488858386498, 10.36176523049612777936358788848, 11.46887574888132052857326460478, 11.87684639922386496374701280457

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