Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 7-s − 4·11-s + 2·13-s + 6·17-s + 4·19-s − 25-s − 2·29-s − 2·35-s − 6·37-s − 2·41-s − 4·43-s + 49-s + 6·53-s + 8·55-s − 12·59-s + 2·61-s − 4·65-s + 4·67-s − 6·73-s − 4·77-s + 16·79-s + 12·83-s − 12·85-s + 14·89-s + 2·91-s − 8·95-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s − 1/5·25-s − 0.371·29-s − 0.338·35-s − 0.986·37-s − 0.312·41-s − 0.609·43-s + 1/7·49-s + 0.824·53-s + 1.07·55-s − 1.56·59-s + 0.256·61-s − 0.496·65-s + 0.488·67-s − 0.702·73-s − 0.455·77-s + 1.80·79-s + 1.31·83-s − 1.30·85-s + 1.48·89-s + 0.209·91-s − 0.820·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 4032,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.473336744\)
\(L(\frac12)\)  \(\approx\)  \(1.473336744\)
\(L(\frac{3}{2})\)   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 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−8.229992153219430708166678625388, −7.70691183501011689426667743544, −7.32441936749744810989380005535, −6.14521652142582152085553341069, −5.34641745697560833577710874285, −4.80722452268725035282926398957, −3.62256199043071801163653681899, −3.22746506277116131252491518684, −1.93480858488759200478154145238, −0.69577924383102779401083388688, 0.69577924383102779401083388688, 1.93480858488759200478154145238, 3.22746506277116131252491518684, 3.62256199043071801163653681899, 4.80722452268725035282926398957, 5.34641745697560833577710874285, 6.14521652142582152085553341069, 7.32441936749744810989380005535, 7.70691183501011689426667743544, 8.229992153219430708166678625388

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