Properties

Degree 2
Conductor $ 2^{3} $
Sign $1$
Motivic weight 20
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.02e3·2-s + 1.14e5·3-s + 1.04e6·4-s + 1.16e8·6-s + 1.07e9·8-s + 9.56e9·9-s − 4.23e10·11-s + 1.19e11·12-s + 1.09e12·16-s − 3.35e12·17-s + 9.79e12·18-s − 1.01e12·19-s − 4.34e13·22-s + 1.22e14·24-s + 9.53e13·25-s + 6.93e14·27-s + 1.12e15·32-s − 4.84e15·33-s − 3.43e15·34-s + 1.00e16·36-s − 1.03e15·38-s − 2.54e16·41-s + 2.78e15·43-s − 4.44e16·44-s + 1.25e17·48-s + 7.97e16·49-s + 9.76e16·50-s + ⋯
L(s)  = 1  + 2-s + 1.93·3-s + 4-s + 1.93·6-s + 8-s + 2.74·9-s − 1.63·11-s + 1.93·12-s + 16-s − 1.66·17-s + 2.74·18-s − 0.165·19-s − 1.63·22-s + 1.93·24-s + 25-s + 3.36·27-s + 32-s − 3.16·33-s − 1.66·34-s + 2.74·36-s − 0.165·38-s − 1.89·41-s + 0.128·43-s − 1.63·44-s + 1.93·48-s + 49-s + 50-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(21-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(20\)
character  :  $\chi_{8} (3, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :10),\ 1)$
$L(\frac{21}{2})$  $\approx$  $6.38528$
$L(\frac12)$  $\approx$  $6.38528$
$L(11)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - p^{10} T \)
good3 \( 1 - 114226 T + p^{20} T^{2} \)
5 \( ( 1 - p^{10} T )( 1 + p^{10} T ) \)
7 \( ( 1 - p^{10} T )( 1 + p^{10} T ) \)
11 \( 1 + 42383023726 T + p^{20} T^{2} \)
13 \( ( 1 - p^{10} T )( 1 + p^{10} T ) \)
17 \( 1 + 3353535763774 T + p^{20} T^{2} \)
19 \( 1 + 1014654432526 T + p^{20} T^{2} \)
23 \( ( 1 - p^{10} T )( 1 + p^{10} T ) \)
29 \( ( 1 - p^{10} T )( 1 + p^{10} T ) \)
31 \( ( 1 - p^{10} T )( 1 + p^{10} T ) \)
37 \( ( 1 - p^{10} T )( 1 + p^{10} T ) \)
41 \( 1 + 25418071370591326 T + p^{20} T^{2} \)
43 \( 1 - 2781113986388498 T + p^{20} T^{2} \)
47 \( ( 1 - p^{10} T )( 1 + p^{10} T ) \)
53 \( ( 1 - p^{10} T )( 1 + p^{10} T ) \)
59 \( 1 + 173912197184497198 T + p^{20} T^{2} \)
61 \( ( 1 - p^{10} T )( 1 + p^{10} T ) \)
67 \( 1 + 356137514166464974 T + p^{20} T^{2} \)
71 \( ( 1 - p^{10} T )( 1 + p^{10} T ) \)
73 \( 1 + 6016717170316692574 T + p^{20} T^{2} \)
79 \( ( 1 - p^{10} T )( 1 + p^{10} T ) \)
83 \( 1 + 31022856480301602574 T + p^{20} T^{2} \)
89 \( 1 - 61202446863210984674 T + p^{20} T^{2} \)
97 \( 1 + 50009130514058267902 T + p^{20} T^{2} \)
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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

−15.79940195485861624712976432897, −15.03012238398468829382782711615, −13.64000556972193717297830740544, −12.92389809175974000285833457325, −10.43912973810416712728046572522, −8.505732636221698784273241567536, −7.14688562005276719945333114426, −4.59021675822229857157808834383, −3.03945142575726488506881529048, −2.07479144646133450674174069486, 2.07479144646133450674174069486, 3.03945142575726488506881529048, 4.59021675822229857157808834383, 7.14688562005276719945333114426, 8.505732636221698784273241567536, 10.43912973810416712728046572522, 12.92389809175974000285833457325, 13.64000556972193717297830740544, 15.03012238398468829382782711615, 15.79940195485861624712976432897

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