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Properties

Label 2-2e8-8.5-c7-0-36
Degree $2$
Conductor $256$
Sign $0.707 + 0.707i$
Analytic cond. $79.9705$
Root an. cond. $8.94262$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 51.2i·3-s − 145. i·5-s + 192.·7-s − 436.·9-s + 5.08e3i·11-s − 9.38e3i·13-s + 7.45e3·15-s − 1.72e4·17-s − 1.72e4i·19-s + 9.86e3i·21-s − 1.84e4·23-s + 5.69e4·25-s + 8.96e4i·27-s − 5.83e4i·29-s + 9.99e4·31-s + ⋯
L(s)  = 1  + 1.09i·3-s − 0.520i·5-s + 0.212·7-s − 0.199·9-s + 1.15i·11-s − 1.18i·13-s + 0.570·15-s − 0.850·17-s − 0.576i·19-s + 0.232i·21-s − 0.315·23-s + 0.729·25-s + 0.876i·27-s − 0.444i·29-s + 0.602·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(79.9705\)
Root analytic conductor: \(8.94262\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :7/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.545204448\)
\(L(\frac12)\) \(\approx\) \(1.545204448\)
\(L(\frac{9}{2})\) 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 \( 1 - 51.2iT - 2.18e3T^{2} \)
5 \( 1 + 145. iT - 7.81e4T^{2} \)
7 \( 1 - 192.T + 8.23e5T^{2} \)
11 \( 1 - 5.08e3iT - 1.94e7T^{2} \)
13 \( 1 + 9.38e3iT - 6.27e7T^{2} \)
17 \( 1 + 1.72e4T + 4.10e8T^{2} \)
19 \( 1 + 1.72e4iT - 8.93e8T^{2} \)
23 \( 1 + 1.84e4T + 3.40e9T^{2} \)
29 \( 1 + 5.83e4iT - 1.72e10T^{2} \)
31 \( 1 - 9.99e4T + 2.75e10T^{2} \)
37 \( 1 + 4.84e5iT - 9.49e10T^{2} \)
41 \( 1 + 5.69e5T + 1.94e11T^{2} \)
43 \( 1 + 1.89e4iT - 2.71e11T^{2} \)
47 \( 1 + 1.22e6T + 5.06e11T^{2} \)
53 \( 1 + 1.64e6iT - 1.17e12T^{2} \)
59 \( 1 - 6.36e4iT - 2.48e12T^{2} \)
61 \( 1 - 1.39e6iT - 3.14e12T^{2} \)
67 \( 1 + 3.02e6iT - 6.06e12T^{2} \)
71 \( 1 - 3.28e6T + 9.09e12T^{2} \)
73 \( 1 - 4.52e6T + 1.10e13T^{2} \)
79 \( 1 - 3.18e6T + 1.92e13T^{2} \)
83 \( 1 + 7.31e6iT - 2.71e13T^{2} \)
89 \( 1 - 3.55e6T + 4.42e13T^{2} \)
97 \( 1 - 1.36e7T + 8.07e13T^{2} \)
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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

−10.46425641993434992957515016494, −9.783482330088016879664998085654, −8.908630560260191472429688987647, −7.87206053608437186432075768634, −6.65086879866943861903912922951, −5.09202545795074136204160759783, −4.66736315935355619991967274055, −3.46160708302512634508231219367, −1.98446851613773914618922771423, −0.38360247149693647106456799754, 1.08582422501984975740835481330, 2.07861701848998877980596651683, 3.35259533748187035290252071211, 4.77585924286215785951743551201, 6.40306981757843698793534263712, 6.67693958469723909224757630044, 7.942083595262209447380271136330, 8.703489033049976661498040452320, 9.996879103232799942035798553888, 11.15101013031980268602333351946

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