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  − 0.297·2-s − 1.91·4-s − 3.06·5-s − 7-s + 1.16·8-s + 0.911·10-s − 5.38·13-s + 0.297·14-s + 3.47·16-s − 6.72·17-s + 0.434·19-s + 5.85·20-s + 2.92·23-s + 4.38·25-s + 1.60·26-s + 1.91·28-s + 6.86·29-s + 4.95·31-s − 3.36·32-s + 2.00·34-s + 3.06·35-s − 3.38·37-s − 0.129·38-s − 3.56·40-s + 9.92·41-s + 1.82·43-s − 0.869·46-s + ⋯
L(s)  = 1  − 0.210·2-s − 0.955·4-s − 1.37·5-s − 0.377·7-s + 0.411·8-s + 0.288·10-s − 1.49·13-s + 0.0795·14-s + 0.869·16-s − 1.63·17-s + 0.0997·19-s + 1.30·20-s + 0.609·23-s + 0.877·25-s + 0.314·26-s + 0.361·28-s + 1.27·29-s + 0.889·31-s − 0.594·32-s + 0.342·34-s + 0.517·35-s − 0.557·37-s − 0.0209·38-s − 0.563·40-s + 1.55·41-s + 0.278·43-s − 0.128·46-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 \( 1 + 0.297T + 2T^{2} \)
5 \( 1 + 3.06T + 5T^{2} \)
13 \( 1 + 5.38T + 13T^{2} \)
17 \( 1 + 6.72T + 17T^{2} \)
19 \( 1 - 0.434T + 19T^{2} \)
23 \( 1 - 2.92T + 23T^{2} \)
29 \( 1 - 6.86T + 29T^{2} \)
31 \( 1 - 4.95T + 31T^{2} \)
37 \( 1 + 3.38T + 37T^{2} \)
41 \( 1 - 9.92T + 41T^{2} \)
43 \( 1 - 1.82T + 43T^{2} \)
47 \( 1 - 7.45T + 47T^{2} \)
53 \( 1 + 9.33T + 53T^{2} \)
59 \( 1 + 7.17T + 59T^{2} \)
61 \( 1 - 6.77T + 61T^{2} \)
67 \( 1 - 3.56T + 67T^{2} \)
71 \( 1 - 8.50T + 71T^{2} \)
73 \( 1 + 2.61T + 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + 1.73T + 83T^{2} \)
89 \( 1 - 7.31T + 89T^{2} \)
97 \( 1 + 6.95T + 97T^{2} \)
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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

−7.59154765621060122581168097836, −7.05279788936799735896681838345, −6.26794461206643767131419742869, −5.12303957449565402629546407681, −4.52552683771378797738389243246, −4.18308472774341442118880275701, −3.18516327942281102377470237241, −2.40251344406843557970290076524, −0.805223434286125553378838092839, 0, 0.805223434286125553378838092839, 2.40251344406843557970290076524, 3.18516327942281102377470237241, 4.18308472774341442118880275701, 4.52552683771378797738389243246, 5.12303957449565402629546407681, 6.26794461206643767131419742869, 7.05279788936799735896681838345, 7.59154765621060122581168097836

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