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  + 1.61·2-s + 0.618·4-s − 5-s − 7-s − 2.23·8-s − 1.61·10-s + 5.47·13-s − 1.61·14-s − 4.85·16-s + 0.763·17-s − 6.70·19-s − 0.618·20-s + 7.70·23-s − 4·25-s + 8.85·26-s − 0.618·28-s + 5·29-s − 0.763·31-s − 3.38·32-s + 1.23·34-s + 35-s − 7·37-s − 10.8·38-s + 2.23·40-s + 6.47·41-s + 7.70·43-s + 12.4·46-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.309·4-s − 0.447·5-s − 0.377·7-s − 0.790·8-s − 0.511·10-s + 1.51·13-s − 0.432·14-s − 1.21·16-s + 0.185·17-s − 1.53·19-s − 0.138·20-s + 1.60·23-s − 0.800·25-s + 1.73·26-s − 0.116·28-s + 0.928·29-s − 0.137·31-s − 0.597·32-s + 0.211·34-s + 0.169·35-s − 1.15·37-s − 1.76·38-s + 0.353·40-s + 1.01·41-s + 1.17·43-s + 1.83·46-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 \( 1 - 1.61T + 2T^{2} \)
5 \( 1 + T + 5T^{2} \)
13 \( 1 - 5.47T + 13T^{2} \)
17 \( 1 - 0.763T + 17T^{2} \)
19 \( 1 + 6.70T + 19T^{2} \)
23 \( 1 - 7.70T + 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 0.763T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 - 6.47T + 41T^{2} \)
43 \( 1 - 7.70T + 43T^{2} \)
47 \( 1 - 4.23T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 14.2T + 67T^{2} \)
71 \( 1 + 6.47T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 - 5.52T + 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 + 3.70T + 97T^{2} \)
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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

−7.40031438710040045426745298275, −6.44482349559888624452860590815, −6.17496929889396809620409417684, −5.39494460446182717767291438229, −4.47338777330616396383442114690, −4.06591202214995441206036166884, −3.29604547639311874100633429578, −2.67357873709595706672890516908, −1.34619014789795605092339838172, 0, 1.34619014789795605092339838172, 2.67357873709595706672890516908, 3.29604547639311874100633429578, 4.06591202214995441206036166884, 4.47338777330616396383442114690, 5.39494460446182717767291438229, 6.17496929889396809620409417684, 6.44482349559888624452860590815, 7.40031438710040045426745298275

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