Properties

Label 2-2e8-1.1-c7-0-37
Degree $2$
Conductor $256$
Sign $1$
Analytic cond. $79.9705$
Root an. cond. $8.94262$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 76.9·3-s + 338.·5-s + 438.·7-s + 3.73e3·9-s + 1.96e3·11-s + 2.21e3·13-s + 2.60e4·15-s − 1.21e4·17-s + 3.28e4·19-s + 3.37e4·21-s − 1.96e4·23-s + 3.64e4·25-s + 1.19e5·27-s + 1.60e5·29-s − 2.29e5·31-s + 1.51e5·33-s + 1.48e5·35-s + 4.96e5·37-s + 1.70e5·39-s − 5.99e5·41-s − 8.83e4·43-s + 1.26e6·45-s + 8.20e5·47-s − 6.30e5·49-s − 9.32e5·51-s + 1.53e6·53-s + 6.65e5·55-s + ⋯
L(s)  = 1  + 1.64·3-s + 1.21·5-s + 0.483·7-s + 1.70·9-s + 0.445·11-s + 0.279·13-s + 1.99·15-s − 0.598·17-s + 1.09·19-s + 0.795·21-s − 0.335·23-s + 0.466·25-s + 1.16·27-s + 1.22·29-s − 1.38·31-s + 0.733·33-s + 0.585·35-s + 1.61·37-s + 0.459·39-s − 1.35·41-s − 0.169·43-s + 2.06·45-s + 1.15·47-s − 0.765·49-s − 0.984·51-s + 1.41·53-s + 0.539·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $1$
Analytic conductor: \(79.9705\)
Root analytic conductor: \(8.94262\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(5.955423966\)
\(L(\frac12)\) \(\approx\) \(5.955423966\)
\(L(\frac{9}{2})\) 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 \( 1 - 76.9T + 2.18e3T^{2} \)
5 \( 1 - 338.T + 7.81e4T^{2} \)
7 \( 1 - 438.T + 8.23e5T^{2} \)
11 \( 1 - 1.96e3T + 1.94e7T^{2} \)
13 \( 1 - 2.21e3T + 6.27e7T^{2} \)
17 \( 1 + 1.21e4T + 4.10e8T^{2} \)
19 \( 1 - 3.28e4T + 8.93e8T^{2} \)
23 \( 1 + 1.96e4T + 3.40e9T^{2} \)
29 \( 1 - 1.60e5T + 1.72e10T^{2} \)
31 \( 1 + 2.29e5T + 2.75e10T^{2} \)
37 \( 1 - 4.96e5T + 9.49e10T^{2} \)
41 \( 1 + 5.99e5T + 1.94e11T^{2} \)
43 \( 1 + 8.83e4T + 2.71e11T^{2} \)
47 \( 1 - 8.20e5T + 5.06e11T^{2} \)
53 \( 1 - 1.53e6T + 1.17e12T^{2} \)
59 \( 1 + 1.82e6T + 2.48e12T^{2} \)
61 \( 1 + 4.84e5T + 3.14e12T^{2} \)
67 \( 1 + 7.98e4T + 6.06e12T^{2} \)
71 \( 1 + 1.27e6T + 9.09e12T^{2} \)
73 \( 1 + 3.70e6T + 1.10e13T^{2} \)
79 \( 1 + 2.55e6T + 1.92e13T^{2} \)
83 \( 1 - 1.53e6T + 2.71e13T^{2} \)
89 \( 1 + 1.99e6T + 4.42e13T^{2} \)
97 \( 1 + 2.89e4T + 8.07e13T^{2} \)
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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

−10.43064246784086935917712844034, −9.538537024918842311210646911878, −8.959580752256968093764760835509, −8.045881679337585051580468987806, −6.99960194537675909588012150951, −5.75242117588184067230339589591, −4.38567292015881479544929675241, −3.15276760577261726138176723117, −2.14490479486028363601341096421, −1.32195462733855491622249957308, 1.32195462733855491622249957308, 2.14490479486028363601341096421, 3.15276760577261726138176723117, 4.38567292015881479544929675241, 5.75242117588184067230339589591, 6.99960194537675909588012150951, 8.045881679337585051580468987806, 8.959580752256968093764760835509, 9.538537024918842311210646911878, 10.43064246784086935917712844034

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