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  − 1.27·2-s − 0.370·4-s + 4.09·5-s + 7-s + 3.02·8-s − 5.22·10-s − 4.39·13-s − 1.27·14-s − 3.12·16-s − 4.19·17-s − 1.24·19-s − 1.51·20-s − 4.97·23-s + 11.7·25-s + 5.60·26-s − 0.370·28-s + 1.93·29-s − 1.56·31-s − 2.06·32-s + 5.35·34-s + 4.09·35-s − 0.716·37-s + 1.59·38-s + 12.3·40-s − 4.80·41-s + 1.35·43-s + 6.34·46-s + ⋯
L(s)  = 1  − 0.902·2-s − 0.185·4-s + 1.82·5-s + 0.377·7-s + 1.06·8-s − 1.65·10-s − 1.21·13-s − 0.341·14-s − 0.780·16-s − 1.01·17-s − 0.286·19-s − 0.338·20-s − 1.03·23-s + 2.34·25-s + 1.09·26-s − 0.0699·28-s + 0.359·29-s − 0.280·31-s − 0.365·32-s + 0.917·34-s + 0.691·35-s − 0.117·37-s + 0.258·38-s + 1.95·40-s − 0.750·41-s + 0.206·43-s + 0.935·46-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 \( 1 + 1.27T + 2T^{2} \)
5 \( 1 - 4.09T + 5T^{2} \)
13 \( 1 + 4.39T + 13T^{2} \)
17 \( 1 + 4.19T + 17T^{2} \)
19 \( 1 + 1.24T + 19T^{2} \)
23 \( 1 + 4.97T + 23T^{2} \)
29 \( 1 - 1.93T + 29T^{2} \)
31 \( 1 + 1.56T + 31T^{2} \)
37 \( 1 + 0.716T + 37T^{2} \)
41 \( 1 + 4.80T + 41T^{2} \)
43 \( 1 - 1.35T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 - 3.97T + 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 - 7.59T + 67T^{2} \)
71 \( 1 + 0.218T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 + 4.56T + 79T^{2} \)
83 \( 1 + 2.45T + 83T^{2} \)
89 \( 1 + 4.20T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−7.61175036350009876855756988457, −6.90589416171391550475565716090, −6.21558970850372994411109306027, −5.41025646114984601712550156300, −4.83823846814469640937408319538, −4.12802665624847346262170888828, −2.62127007530647430944853052976, −2.07262487854928013845830593767, −1.34505756331115863243341067528, 0, 1.34505756331115863243341067528, 2.07262487854928013845830593767, 2.62127007530647430944853052976, 4.12802665624847346262170888828, 4.83823846814469640937408319538, 5.41025646114984601712550156300, 6.21558970850372994411109306027, 6.90589416171391550475565716090, 7.61175036350009876855756988457

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