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  − 2.23·2-s + 3.00·4-s + 0.618·5-s + 7-s − 2.23·8-s − 1.38·10-s + 3.23·13-s − 2.23·14-s − 0.999·16-s − 4.85·17-s + 2.85·19-s + 1.85·20-s − 4.38·23-s − 4.61·25-s − 7.23·26-s + 3.00·28-s + 6·29-s − 3.09·31-s + 6.70·32-s + 10.8·34-s + 0.618·35-s + 4.61·37-s − 6.38·38-s − 1.38·40-s − 7.38·41-s + 9.70·43-s + 9.79·46-s + ⋯
L(s)  = 1  − 1.58·2-s + 1.50·4-s + 0.276·5-s + 0.377·7-s − 0.790·8-s − 0.437·10-s + 0.897·13-s − 0.597·14-s − 0.249·16-s − 1.17·17-s + 0.654·19-s + 0.414·20-s − 0.913·23-s − 0.923·25-s − 1.41·26-s + 0.566·28-s + 1.11·29-s − 0.555·31-s + 1.18·32-s + 1.86·34-s + 0.104·35-s + 0.759·37-s − 1.03·38-s − 0.218·40-s − 1.15·41-s + 1.48·43-s + 1.44·46-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 \( 1 + 2.23T + 2T^{2} \)
5 \( 1 - 0.618T + 5T^{2} \)
13 \( 1 - 3.23T + 13T^{2} \)
17 \( 1 + 4.85T + 17T^{2} \)
19 \( 1 - 2.85T + 19T^{2} \)
23 \( 1 + 4.38T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 3.09T + 31T^{2} \)
37 \( 1 - 4.61T + 37T^{2} \)
41 \( 1 + 7.38T + 41T^{2} \)
43 \( 1 - 9.70T + 43T^{2} \)
47 \( 1 + 4.47T + 47T^{2} \)
53 \( 1 + 6.76T + 53T^{2} \)
59 \( 1 + 3.23T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 4.94T + 71T^{2} \)
73 \( 1 - 13.7T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 + 5.61T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−7.82504366911934375959338371316, −6.99637478132851251699325244865, −6.38199236330212949739430568276, −5.70941443819596765643991295399, −4.66343163439297329573540439333, −3.89891644587520754339340644751, −2.72230315961593510172873749443, −1.90173726130557904002891758693, −1.18866396551325227321858583292, 0, 1.18866396551325227321858583292, 1.90173726130557904002891758693, 2.72230315961593510172873749443, 3.89891644587520754339340644751, 4.66343163439297329573540439333, 5.70941443819596765643991295399, 6.38199236330212949739430568276, 6.99637478132851251699325244865, 7.82504366911934375959338371316

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