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  + 2.41·2-s + 3.84·4-s − 4.01·5-s − 7-s + 4.45·8-s − 9.69·10-s + 0.690·13-s − 2.41·14-s + 3.08·16-s + 5.78·17-s + 1.80·19-s − 15.4·20-s − 2.03·23-s + 11.0·25-s + 1.66·26-s − 3.84·28-s + 7.98·29-s − 10.4·31-s − 1.46·32-s + 13.9·34-s + 4.01·35-s − 7.44·37-s + 4.35·38-s − 17.8·40-s − 5.66·41-s − 3.44·43-s − 4.92·46-s + ⋯
L(s)  = 1  + 1.70·2-s + 1.92·4-s − 1.79·5-s − 0.377·7-s + 1.57·8-s − 3.06·10-s + 0.191·13-s − 0.646·14-s + 0.770·16-s + 1.40·17-s + 0.413·19-s − 3.44·20-s − 0.424·23-s + 2.21·25-s + 0.327·26-s − 0.726·28-s + 1.48·29-s − 1.88·31-s − 0.258·32-s + 2.39·34-s + 0.677·35-s − 1.22·37-s + 0.706·38-s − 2.82·40-s − 0.884·41-s − 0.524·43-s − 0.725·46-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 \( 1 - 2.41T + 2T^{2} \)
5 \( 1 + 4.01T + 5T^{2} \)
13 \( 1 - 0.690T + 13T^{2} \)
17 \( 1 - 5.78T + 17T^{2} \)
19 \( 1 - 1.80T + 19T^{2} \)
23 \( 1 + 2.03T + 23T^{2} \)
29 \( 1 - 7.98T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + 7.44T + 37T^{2} \)
41 \( 1 + 5.66T + 41T^{2} \)
43 \( 1 + 3.44T + 43T^{2} \)
47 \( 1 + 7.32T + 47T^{2} \)
53 \( 1 - 1.79T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 - 6.95T + 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 7.20T + 73T^{2} \)
79 \( 1 - 5.33T + 79T^{2} \)
83 \( 1 + 8.59T + 83T^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 + 9.50T + 97T^{2} \)
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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

−7.19072519875889517835821793098, −6.90549609971924815367835406509, −5.94574377233578203150833060565, −5.22196872586893273877593018896, −4.64243110854830405663066199653, −3.78046570291997258509867455951, −3.44793149633773440059774665592, −2.89003615833558242830249029789, −1.47793753599588380973586044960, 0, 1.47793753599588380973586044960, 2.89003615833558242830249029789, 3.44793149633773440059774665592, 3.78046570291997258509867455951, 4.64243110854830405663066199653, 5.22196872586893273877593018896, 5.94574377233578203150833060565, 6.90549609971924815367835406509, 7.19072519875889517835821793098

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