Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 9-s + 2·11-s − 2·15-s − 2·17-s + 2·23-s − 25-s − 27-s − 6·29-s + 4·31-s − 2·33-s − 6·37-s + 2·41-s + 2·45-s + 2·51-s + 6·53-s + 4·55-s + 12·59-s + 12·61-s + 12·67-s − 2·69-s − 10·71-s − 12·73-s + 75-s + 12·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.603·11-s − 0.516·15-s − 0.485·17-s + 0.417·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.348·33-s − 0.986·37-s + 0.312·41-s + 0.298·45-s + 0.280·51-s + 0.824·53-s + 0.539·55-s + 1.56·59-s + 1.53·61-s + 1.46·67-s − 0.240·69-s − 1.18·71-s − 1.40·73-s + 0.115·75-s + 1.35·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9408\)    =    \(2^{6} \cdot 3 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{9408} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 9408,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.009654054\)
\(L(\frac12)\)  \(\approx\)  \(2.009654054\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−7.52881062080287534363231413476, −6.85867568713600386564320173881, −6.35107651136982918962627237948, −5.61536212029128238744767789341, −5.15687836404312544898113092853, −4.23441465160980836258516474733, −3.55658849077076747446536169521, −2.40632386661151840577141944197, −1.73944429407055584432017355856, −0.71012541302141019601397423071, 0.71012541302141019601397423071, 1.73944429407055584432017355856, 2.40632386661151840577141944197, 3.55658849077076747446536169521, 4.23441465160980836258516474733, 5.15687836404312544898113092853, 5.61536212029128238744767789341, 6.35107651136982918962627237948, 6.85867568713600386564320173881, 7.52881062080287534363231413476

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