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  − 2.55·2-s + 4.51·4-s − 3.42·5-s + 7-s − 6.41·8-s + 8.75·10-s − 2.17·13-s − 2.55·14-s + 7.33·16-s + 4.51·17-s − 2.42·19-s − 15.4·20-s − 0.648·23-s + 6.75·25-s + 5.55·26-s + 4.51·28-s + 1.25·29-s − 8.03·31-s − 5.90·32-s − 11.5·34-s − 3.42·35-s + 5.01·37-s + 6.18·38-s + 21.9·40-s + 2.62·41-s − 1.46·43-s + 1.65·46-s + ⋯
L(s)  = 1  − 1.80·2-s + 2.25·4-s − 1.53·5-s + 0.377·7-s − 2.26·8-s + 2.76·10-s − 0.603·13-s − 0.682·14-s + 1.83·16-s + 1.09·17-s − 0.556·19-s − 3.45·20-s − 0.135·23-s + 1.35·25-s + 1.08·26-s + 0.852·28-s + 0.232·29-s − 1.44·31-s − 1.04·32-s − 1.97·34-s − 0.579·35-s + 0.824·37-s + 1.00·38-s + 3.47·40-s + 0.409·41-s − 0.222·43-s + 0.243·46-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 \( 1 + 2.55T + 2T^{2} \)
5 \( 1 + 3.42T + 5T^{2} \)
13 \( 1 + 2.17T + 13T^{2} \)
17 \( 1 - 4.51T + 17T^{2} \)
19 \( 1 + 2.42T + 19T^{2} \)
23 \( 1 + 0.648T + 23T^{2} \)
29 \( 1 - 1.25T + 29T^{2} \)
31 \( 1 + 8.03T + 31T^{2} \)
37 \( 1 - 5.01T + 37T^{2} \)
41 \( 1 - 2.62T + 41T^{2} \)
43 \( 1 + 1.46T + 43T^{2} \)
47 \( 1 - 5.04T + 47T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 + 7.66T + 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 - 6.22T + 67T^{2} \)
71 \( 1 + 4.22T + 71T^{2} \)
73 \( 1 + 5.92T + 73T^{2} \)
79 \( 1 - 9.76T + 79T^{2} \)
83 \( 1 - 8.37T + 83T^{2} \)
89 \( 1 + 4.76T + 89T^{2} \)
97 \( 1 + 8.70T + 97T^{2} \)
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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

−7.79607180428168816343905489156, −7.26846772917002783082955672082, −6.58167633854256035084133550522, −5.61110431322419641011122381689, −4.61395307873377013212955991821, −3.74496763312182612608797484862, −2.91002804611702792917356377945, −1.93544757907957635512886099955, −0.896429490626231175202944561293, 0, 0.896429490626231175202944561293, 1.93544757907957635512886099955, 2.91002804611702792917356377945, 3.74496763312182612608797484862, 4.61395307873377013212955991821, 5.61110431322419641011122381689, 6.58167633854256035084133550522, 7.26846772917002783082955672082, 7.79607180428168816343905489156

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