Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

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 − 6·17-s − 2·19-s + 6·20-s − 3·23-s + 4·25-s + 2·28-s − 6·29-s + 5·31-s + 3·35-s + 11·37-s + 6·41-s − 8·43-s + 49-s − 8·52-s + 6·53-s + 9·59-s + 10·61-s − 8·64-s − 12·65-s + 5·67-s + 12·68-s − 9·71-s + ⋯
L(s)  = 1  − 4-s − 1.34·5-s − 0.377·7-s + 1.10·13-s + 16-s − 1.45·17-s − 0.458·19-s + 1.34·20-s − 0.625·23-s + 4/5·25-s + 0.377·28-s − 1.11·29-s + 0.898·31-s + 0.507·35-s + 1.80·37-s + 0.937·41-s − 1.21·43-s + 1/7·49-s − 1.10·52-s + 0.824·53-s + 1.17·59-s + 1.28·61-s − 64-s − 1.48·65-s + 0.610·67-s + 1.45·68-s − 1.06·71-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  :  $\chi_{7623} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−17.57528172087005, −16.48596844060430, −16.30918060910241, −15.53065855431893, −15.05375226577759, −14.56405984795201, −13.57805937267101, −13.25628542850248, −12.79847865361677, −11.94879242633959, −11.40866750887114, −10.89227678330219, −10.11536981365809, −9.356157357685715, −8.783353013610332, −8.198299351340585, −7.797801899330628, −6.810628542343045, −6.199543555600814, −5.350594301662208, −4.312977329894808, −4.099812815618040, −3.451314804871798, −2.335321503941461, −0.8968101269363714, 0, 0.8968101269363714, 2.335321503941461, 3.451314804871798, 4.099812815618040, 4.312977329894808, 5.350594301662208, 6.199543555600814, 6.810628542343045, 7.797801899330628, 8.198299351340585, 8.783353013610332, 9.356157357685715, 10.11536981365809, 10.89227678330219, 11.40866750887114, 11.94879242633959, 12.79847865361677, 13.25628542850248, 13.57805937267101, 14.56405984795201, 15.05375226577759, 15.53065855431893, 16.30918060910241, 16.48596844060430, 17.57528172087005

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