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  + 1.38·2-s − 0.0791·4-s + 0.133·5-s + 7-s − 2.88·8-s + 0.184·10-s − 0.641·13-s + 1.38·14-s − 3.83·16-s + 1.42·17-s + 7.18·19-s − 0.0105·20-s − 1.66·23-s − 4.98·25-s − 0.889·26-s − 0.0791·28-s − 4.47·29-s − 6.83·31-s + 0.447·32-s + 1.97·34-s + 0.133·35-s + 3.76·37-s + 9.95·38-s − 0.383·40-s − 6.17·41-s − 1.03·43-s − 2.30·46-s + ⋯
L(s)  = 1  + 0.980·2-s − 0.0395·4-s + 0.0594·5-s + 0.377·7-s − 1.01·8-s + 0.0582·10-s − 0.177·13-s + 0.370·14-s − 0.958·16-s + 0.345·17-s + 1.64·19-s − 0.00235·20-s − 0.347·23-s − 0.996·25-s − 0.174·26-s − 0.0149·28-s − 0.831·29-s − 1.22·31-s + 0.0790·32-s + 0.338·34-s + 0.0224·35-s + 0.619·37-s + 1.61·38-s − 0.0605·40-s − 0.964·41-s − 0.158·43-s − 0.340·46-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 \( 1 - 1.38T + 2T^{2} \)
5 \( 1 - 0.133T + 5T^{2} \)
13 \( 1 + 0.641T + 13T^{2} \)
17 \( 1 - 1.42T + 17T^{2} \)
19 \( 1 - 7.18T + 19T^{2} \)
23 \( 1 + 1.66T + 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + 6.83T + 31T^{2} \)
37 \( 1 - 3.76T + 37T^{2} \)
41 \( 1 + 6.17T + 41T^{2} \)
43 \( 1 + 1.03T + 43T^{2} \)
47 \( 1 + 9.48T + 47T^{2} \)
53 \( 1 - 0.666T + 53T^{2} \)
59 \( 1 + 8.18T + 59T^{2} \)
61 \( 1 - 9.49T + 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + 4.83T + 71T^{2} \)
73 \( 1 + 8.85T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + 9.57T + 83T^{2} \)
89 \( 1 - 17.7T + 89T^{2} \)
97 \( 1 - 6.58T + 97T^{2} \)
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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

−7.54890511395329086858834385663, −6.69253910263887259571381251042, −5.84964113521415545085051892127, −5.32648414635007633991834739598, −4.84104345560798209090117224803, −3.81176263448534691824252059644, −3.45691980963990039406963298267, −2.44757993034458400259998902523, −1.41829130552864850027875300460, 0, 1.41829130552864850027875300460, 2.44757993034458400259998902523, 3.45691980963990039406963298267, 3.81176263448534691824252059644, 4.84104345560798209090117224803, 5.32648414635007633991834739598, 5.84964113521415545085051892127, 6.69253910263887259571381251042, 7.54890511395329086858834385663

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