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  − 0.120·2-s − 1.98·4-s + 2.80·5-s + 7-s + 0.479·8-s − 0.338·10-s + 1.07·13-s − 0.120·14-s + 3.91·16-s − 6.95·17-s + 7.54·19-s − 5.57·20-s − 4.82·23-s + 2.88·25-s − 0.129·26-s − 1.98·28-s − 1.22·29-s − 8.07·31-s − 1.43·32-s + 0.837·34-s + 2.80·35-s + 1.53·37-s − 0.908·38-s + 1.34·40-s − 9.29·41-s − 5.23·43-s + 0.580·46-s + ⋯
L(s)  = 1  − 0.0851·2-s − 0.992·4-s + 1.25·5-s + 0.377·7-s + 0.169·8-s − 0.106·10-s + 0.299·13-s − 0.0321·14-s + 0.978·16-s − 1.68·17-s + 1.73·19-s − 1.24·20-s − 1.00·23-s + 0.577·25-s − 0.0254·26-s − 0.375·28-s − 0.227·29-s − 1.45·31-s − 0.252·32-s + 0.143·34-s + 0.474·35-s + 0.252·37-s − 0.147·38-s + 0.213·40-s − 1.45·41-s − 0.798·43-s + 0.0856·46-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 \( 1 + 0.120T + 2T^{2} \)
5 \( 1 - 2.80T + 5T^{2} \)
13 \( 1 - 1.07T + 13T^{2} \)
17 \( 1 + 6.95T + 17T^{2} \)
19 \( 1 - 7.54T + 19T^{2} \)
23 \( 1 + 4.82T + 23T^{2} \)
29 \( 1 + 1.22T + 29T^{2} \)
31 \( 1 + 8.07T + 31T^{2} \)
37 \( 1 - 1.53T + 37T^{2} \)
41 \( 1 + 9.29T + 41T^{2} \)
43 \( 1 + 5.23T + 43T^{2} \)
47 \( 1 - 1.89T + 47T^{2} \)
53 \( 1 - 3.82T + 53T^{2} \)
59 \( 1 - 6.66T + 59T^{2} \)
61 \( 1 + 9.79T + 61T^{2} \)
67 \( 1 + 2.06T + 67T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 - 2.56T + 73T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 + 2.04T + 83T^{2} \)
89 \( 1 + 4.76T + 89T^{2} \)
97 \( 1 - 9.11T + 97T^{2} \)
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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

−7.55844234111009403578939652955, −6.84191145415349702126946336125, −5.88952850669017651802718492164, −5.49290687833638136313607960267, −4.78461359602230213688087318146, −4.02970108143517780796670012618, −3.15312903198217191779564736761, −2.02841710688677031366616263048, −1.39233925244135548152890474535, 0, 1.39233925244135548152890474535, 2.02841710688677031366616263048, 3.15312903198217191779564736761, 4.02970108143517780796670012618, 4.78461359602230213688087318146, 5.49290687833638136313607960267, 5.88952850669017651802718492164, 6.84191145415349702126946336125, 7.55844234111009403578939652955

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