Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11^{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  − 2·4-s + 3·5-s + 7-s − 4·13-s + 4·16-s + 3·17-s + 2·19-s − 6·20-s + 4·25-s − 2·28-s + 6·29-s + 2·31-s + 3·35-s − 10·37-s − 6·41-s + 11·43-s + 3·47-s + 49-s + 8·52-s − 12·53-s + 3·59-s + 8·61-s − 8·64-s − 12·65-s + 5·67-s − 6·68-s + 12·71-s + ⋯
L(s)  = 1  − 4-s + 1.34·5-s + 0.377·7-s − 1.10·13-s + 16-s + 0.727·17-s + 0.458·19-s − 1.34·20-s + 4/5·25-s − 0.377·28-s + 1.11·29-s + 0.359·31-s + 0.507·35-s − 1.64·37-s − 0.937·41-s + 1.67·43-s + 0.437·47-s + 1/7·49-s + 1.10·52-s − 1.64·53-s + 0.390·59-s + 1.02·61-s − 64-s − 1.48·65-s + 0.610·67-s − 0.727·68-s + 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7623\)    =    \(3^{2} \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7623} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.141285193$
$L(\frac12)$  $\approx$  $2.141285193$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.988558503480528851591159166502, −7.19741603700330141689904385521, −6.40499978060560693859412775955, −5.50903374703667124518338333851, −5.19719441753951767051998124967, −4.53697957520267012817369987236, −3.52343141720749364485357044732, −2.63643283341237998276585249813, −1.74086308734870954921413891975, −0.76208578730909250200409296360, 0.76208578730909250200409296360, 1.74086308734870954921413891975, 2.63643283341237998276585249813, 3.52343141720749364485357044732, 4.53697957520267012817369987236, 5.19719441753951767051998124967, 5.50903374703667124518338333851, 6.40499978060560693859412775955, 7.19741603700330141689904385521, 7.988558503480528851591159166502

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