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  + 1.61·2-s + 0.618·4-s + 4.23·5-s + 7-s − 2.23·8-s + 6.85·10-s − 6.23·13-s + 1.61·14-s − 4.85·16-s − 4.47·17-s − 3·19-s + 2.61·20-s − 8.47·23-s + 12.9·25-s − 10.0·26-s + 0.618·28-s − 3·29-s − 3.38·32-s − 7.23·34-s + 4.23·35-s + 3.47·37-s − 4.85·38-s − 9.47·40-s + 1.52·41-s − 10.9·43-s − 13.7·46-s + 3·47-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.309·4-s + 1.89·5-s + 0.377·7-s − 0.790·8-s + 2.16·10-s − 1.72·13-s + 0.432·14-s − 1.21·16-s − 1.08·17-s − 0.688·19-s + 0.585·20-s − 1.76·23-s + 2.58·25-s − 1.97·26-s + 0.116·28-s − 0.557·29-s − 0.597·32-s − 1.24·34-s + 0.716·35-s + 0.570·37-s − 0.787·38-s − 1.49·40-s + 0.238·41-s − 1.66·43-s − 2.02·46-s + 0.437·47-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  =  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 \( 1 - 1.61T + 2T^{2} \)
5 \( 1 - 4.23T + 5T^{2} \)
13 \( 1 + 6.23T + 13T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 + 8.47T + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 3.47T + 37T^{2} \)
41 \( 1 - 1.52T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + 8.94T + 53T^{2} \)
59 \( 1 - 1.47T + 59T^{2} \)
61 \( 1 + 3.52T + 61T^{2} \)
67 \( 1 + 8.70T + 67T^{2} \)
71 \( 1 + 1.52T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−7.26431258075961080975780352399, −6.40847190316644082491886012821, −6.08890221009689197627026391744, −5.32047121164360914964388329543, −4.77726091023720538350779707239, −4.24557051341291062627371141018, −3.02436133877598197689771950347, −2.22400806894733838308411387173, −1.89095327592551982788316470165, 0, 1.89095327592551982788316470165, 2.22400806894733838308411387173, 3.02436133877598197689771950347, 4.24557051341291062627371141018, 4.77726091023720538350779707239, 5.32047121164360914964388329543, 6.08890221009689197627026391744, 6.40847190316644082491886012821, 7.26431258075961080975780352399

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