Properties

Label 2-2e8-8.5-c5-0-6
Degree $2$
Conductor $256$
Sign $-0.707 - 0.707i$
Analytic cond. $41.0582$
Root an. cond. $6.40767$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 20i·3-s − 74i·5-s − 24·7-s − 157·9-s − 124i·11-s − 478i·13-s + 1.48e3·15-s − 1.19e3·17-s + 3.04e3i·19-s − 480i·21-s + 184·23-s − 2.35e3·25-s + 1.72e3i·27-s + 3.28e3i·29-s + 5.72e3·31-s + ⋯
L(s)  = 1  + 1.28i·3-s − 1.32i·5-s − 0.185·7-s − 0.646·9-s − 0.308i·11-s − 0.784i·13-s + 1.69·15-s − 1.00·17-s + 1.93i·19-s − 0.237i·21-s + 0.0725·23-s − 0.752·25-s + 0.454i·27-s + 0.724i·29-s + 1.07·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(41.0582\)
Root analytic conductor: \(6.40767\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :5/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.092727895\)
\(L(\frac12)\) \(\approx\) \(1.092727895\)
\(L(\frac{7}{2})\) 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 \( 1 - 20iT - 243T^{2} \)
5 \( 1 + 74iT - 3.12e3T^{2} \)
7 \( 1 + 24T + 1.68e4T^{2} \)
11 \( 1 + 124iT - 1.61e5T^{2} \)
13 \( 1 + 478iT - 3.71e5T^{2} \)
17 \( 1 + 1.19e3T + 1.41e6T^{2} \)
19 \( 1 - 3.04e3iT - 2.47e6T^{2} \)
23 \( 1 - 184T + 6.43e6T^{2} \)
29 \( 1 - 3.28e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.72e3T + 2.86e7T^{2} \)
37 \( 1 - 1.03e4iT - 6.93e7T^{2} \)
41 \( 1 - 8.88e3T + 1.15e8T^{2} \)
43 \( 1 - 9.18e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.36e4T + 2.29e8T^{2} \)
53 \( 1 - 1.16e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.68e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.84e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.55e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.19e4T + 1.80e9T^{2} \)
73 \( 1 - 4.88e3T + 2.07e9T^{2} \)
79 \( 1 + 4.45e4T + 3.07e9T^{2} \)
83 \( 1 - 6.73e4iT - 3.93e9T^{2} \)
89 \( 1 + 7.19e4T + 5.58e9T^{2} \)
97 \( 1 - 4.88e4T + 8.58e9T^{2} \)
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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

−11.40307937028845557837432240109, −10.32474364038249819197637538675, −9.685606024860844524266183740564, −8.719924864879738623924814177993, −8.016298864661784869699502929620, −6.19369643979329962808526480953, −5.10600479756497898249165081885, −4.37991032047716966787286406760, −3.26016466579534332080095697165, −1.27978740445730790762136059629, 0.31672458671678899582905607413, 2.00778477468551833986573813147, 2.81664970818251678839883606371, 4.47251497737943694028708211884, 6.28000939179846108232812767170, 6.84726858017668134113999085001, 7.42771071193203155865829207830, 8.748820503442784800962148963092, 9.886029529969304667063226540731, 11.11978352593331766964150968316

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