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-s − 4-s + 2.85·5-s − 7-s + 3·8-s − 2.85·10-s + 1.23·13-s + 14-s − 16-s + 7.85·17-s − 2.61·19-s − 2.85·20-s + 3.09·23-s + 3.14·25-s − 1.23·26-s + 28-s − 2·29-s − 7.32·31-s − 5·32-s − 7.85·34-s − 2.85·35-s − 12.0·37-s + 2.61·38-s + 8.56·40-s − 10.8·41-s + 1.23·43-s − 3.09·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.5·4-s + 1.27·5-s − 0.377·7-s + 1.06·8-s − 0.902·10-s + 0.342·13-s + 0.267·14-s − 0.250·16-s + 1.90·17-s − 0.600·19-s − 0.638·20-s + 0.644·23-s + 0.629·25-s − 0.242·26-s + 0.188·28-s − 0.371·29-s − 1.31·31-s − 0.883·32-s − 1.34·34-s − 0.482·35-s − 1.98·37-s + 0.424·38-s + 1.35·40-s − 1.69·41-s + 0.188·43-s − 0.455·46-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 \( 1 + T + 2T^{2} \)
5 \( 1 - 2.85T + 5T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 - 7.85T + 17T^{2} \)
19 \( 1 + 2.61T + 19T^{2} \)
23 \( 1 - 3.09T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 7.32T + 31T^{2} \)
37 \( 1 + 12.0T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 1.23T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 + 6.76T + 59T^{2} \)
61 \( 1 + 8.94T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 - 5.23T + 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 - 2.94T + 83T^{2} \)
89 \( 1 + 1.09T + 89T^{2} \)
97 \( 1 + 3.52T + 97T^{2} \)
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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

−7.66340059004799438838515002534, −6.88603948178923700006979658522, −6.14632144519866587344601277051, −5.34707741832851384457919340182, −5.02293874384626147573678822029, −3.74501388039750696524107084598, −3.17463509154787482144334220198, −1.83374285155447229927655523625, −1.36174737978857535738788788923, 0, 1.36174737978857535738788788923, 1.83374285155447229927655523625, 3.17463509154787482144334220198, 3.74501388039750696524107084598, 5.02293874384626147573678822029, 5.34707741832851384457919340182, 6.14632144519866587344601277051, 6.88603948178923700006979658522, 7.66340059004799438838515002534

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