Properties

Degree 2
Conductor $ 3^{2} \cdot 43 $
Sign $1$
Motivic weight 2
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s + 21·11-s − 17·13-s + 16·16-s + 9·17-s − 3·23-s + 25·25-s + 19·31-s − 39·41-s − 43·43-s + 84·44-s + 78·47-s + 49·49-s − 68·52-s − 63·53-s + 54·59-s + 64·64-s + 91·67-s + 36·68-s − 14·79-s − 123·83-s − 12·92-s − 193·97-s + 100·100-s − 159·101-s − 181·103-s − 42·107-s + ⋯
L(s)  = 1  + 4-s + 1.90·11-s − 1.30·13-s + 16-s + 9/17·17-s − 0.130·23-s + 25-s + 0.612·31-s − 0.951·41-s − 43-s + 1.90·44-s + 1.65·47-s + 49-s − 1.30·52-s − 1.18·53-s + 0.915·59-s + 64-s + 1.35·67-s + 9/17·68-s − 0.177·79-s − 1.48·83-s − 0.130·92-s − 1.98·97-s + 100-s − 1.57·101-s − 1.75·103-s − 0.392·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(387\)    =    \(3^{2} \cdot 43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  $\chi_{387} (343, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 387,\ (\ :1),\ 1)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.30508\)
\(L(\frac12)\)  \(\approx\)  \(2.30508\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;43\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 + p T \)
good2 \( ( 1 - p T )( 1 + p T ) \)
5 \( ( 1 - p T )( 1 + p T ) \)
7 \( ( 1 - p T )( 1 + p T ) \)
11 \( 1 - 21 T + p^{2} T^{2} \)
13 \( 1 + 17 T + p^{2} T^{2} \)
17 \( 1 - 9 T + p^{2} T^{2} \)
19 \( ( 1 - p T )( 1 + p T ) \)
23 \( 1 + 3 T + p^{2} T^{2} \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 - 19 T + p^{2} T^{2} \)
37 \( ( 1 - p T )( 1 + p T ) \)
41 \( 1 + 39 T + p^{2} T^{2} \)
47 \( 1 - 78 T + p^{2} T^{2} \)
53 \( 1 + 63 T + p^{2} T^{2} \)
59 \( 1 - 54 T + p^{2} T^{2} \)
61 \( ( 1 - p T )( 1 + p T ) \)
67 \( 1 - 91 T + p^{2} T^{2} \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( ( 1 - p T )( 1 + p T ) \)
79 \( 1 + 14 T + p^{2} T^{2} \)
83 \( 1 + 123 T + p^{2} T^{2} \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( 1 + 193 T + p^{2} T^{2} \)
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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

−11.25044150787781939639381284021, −10.19557427297173655848752306076, −9.412508172947311189847837583100, −8.277140290397387096617856577829, −7.07867103715640404421556001380, −6.61545512344545415078049133458, −5.37609669643299440829738158459, −3.99424623108223917905424003768, −2.71989431861053188899707025193, −1.33151835608797037147591353617, 1.33151835608797037147591353617, 2.71989431861053188899707025193, 3.99424623108223917905424003768, 5.37609669643299440829738158459, 6.61545512344545415078049133458, 7.07867103715640404421556001380, 8.277140290397387096617856577829, 9.412508172947311189847837583100, 10.19557427297173655848752306076, 11.25044150787781939639381284021

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