Properties

Degree 2
Conductor $ 2^{3} $
Sign $-1$
Motivic weight 13
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·3-s − 4.33e3·5-s − 1.39e5·7-s − 1.59e6·9-s − 6.48e6·11-s − 2.25e7·13-s + 5.19e4·15-s − 2.37e7·17-s + 3.25e8·19-s + 1.67e6·21-s + 9.21e8·23-s − 1.20e9·25-s + 3.82e7·27-s − 3.86e9·29-s − 2.25e9·31-s + 7.78e7·33-s + 6.06e8·35-s + 1.82e10·37-s + 2.71e8·39-s + 3.44e10·41-s − 1.71e10·43-s + 6.90e9·45-s − 6.73e10·47-s − 7.72e10·49-s + 2.84e8·51-s − 8.72e10·53-s + 2.80e10·55-s + ⋯
L(s)  = 1  − 0.00950·3-s − 0.123·5-s − 0.449·7-s − 0.999·9-s − 1.10·11-s − 1.29·13-s + 0.00117·15-s − 0.238·17-s + 1.58·19-s + 0.00427·21-s + 1.29·23-s − 0.984·25-s + 0.0190·27-s − 1.20·29-s − 0.456·31-s + 0.0104·33-s + 0.0557·35-s + 1.16·37-s + 0.0123·39-s + 1.13·41-s − 0.414·43-s + 0.123·45-s − 0.911·47-s − 0.797·49-s + 0.00226·51-s − 0.540·53-s + 0.136·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(13\)
character  :  $\chi_{8} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8,\ (\ :13/2),\ -1)$
$L(7)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{15}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
good3 \( 1 + 4 p T + p^{13} T^{2} \)
5 \( 1 + 866 p T + p^{13} T^{2} \)
7 \( 1 + 139992 T + p^{13} T^{2} \)
11 \( 1 + 589484 p T + p^{13} T^{2} \)
13 \( 1 + 22588034 T + p^{13} T^{2} \)
17 \( 1 + 23732270 T + p^{13} T^{2} \)
19 \( 1 - 325344836 T + p^{13} T^{2} \)
23 \( 1 - 921600632 T + p^{13} T^{2} \)
29 \( 1 + 3865879218 T + p^{13} T^{2} \)
31 \( 1 + 2253401440 T + p^{13} T^{2} \)
37 \( 1 - 18250384566 T + p^{13} T^{2} \)
41 \( 1 - 34422845322 T + p^{13} T^{2} \)
43 \( 1 + 17192501444 T + p^{13} T^{2} \)
47 \( 1 + 67371749904 T + p^{13} T^{2} \)
53 \( 1 + 1646815442 p T + p^{13} T^{2} \)
59 \( 1 - 540214518668 T + p^{13} T^{2} \)
61 \( 1 + 51276568850 T + p^{13} T^{2} \)
67 \( 1 - 25519930676 T + p^{13} T^{2} \)
71 \( 1 + 1387500699032 T + p^{13} T^{2} \)
73 \( 1 + 819049441238 T + p^{13} T^{2} \)
79 \( 1 + 4030935615344 T + p^{13} T^{2} \)
83 \( 1 - 4180823831428 T + p^{13} T^{2} \)
89 \( 1 - 2677027798266 T + p^{13} T^{2} \)
97 \( 1 + 14039464316446 T + p^{13} T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.74613887921520654007501183408, −16.30843994682926418456907351079, −14.76134913705683884593655838189, −13.10287236351401590976111015832, −11.42242828705978427413891468248, −9.570237317286116045057485229674, −7.58503609122953957763133204661, −5.36986300546103870106011825468, −2.84378142593756197221758683115, 0, 2.84378142593756197221758683115, 5.36986300546103870106011825468, 7.58503609122953957763133204661, 9.570237317286116045057485229674, 11.42242828705978427413891468248, 13.10287236351401590976111015832, 14.76134913705683884593655838189, 16.30843994682926418456907351079, 17.74613887921520654007501183408

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