Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.05·2-s + 2.23·4-s + 1.05·5-s − 7-s − 0.492·8-s − 2.18·10-s + 3.80·13-s + 2.05·14-s − 3.46·16-s + 2.56·17-s − 0.180·19-s + 2.37·20-s − 0.433·23-s − 3.87·25-s − 7.83·26-s − 2.23·28-s + 9.03·29-s − 0.492·31-s + 8.11·32-s − 5.28·34-s − 1.05·35-s − 0.775·37-s + 0.371·38-s − 0.521·40-s + 3.09·41-s − 1.61·43-s + 0.893·46-s + ⋯
L(s)  = 1  − 1.45·2-s + 1.11·4-s + 0.473·5-s − 0.377·7-s − 0.174·8-s − 0.689·10-s + 1.05·13-s + 0.550·14-s − 0.866·16-s + 0.622·17-s − 0.0413·19-s + 0.530·20-s − 0.0904·23-s − 0.775·25-s − 1.53·26-s − 0.423·28-s + 1.67·29-s − 0.0884·31-s + 1.43·32-s − 0.906·34-s − 0.178·35-s − 0.127·37-s + 0.0602·38-s − 0.0825·40-s + 0.483·41-s − 0.246·43-s + 0.131·46-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  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.033037088$
$L(\frac12)$  $\approx$  $1.033037088$
$L(\frac{3}{2})$   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 \( 1 + 2.05T + 2T^{2} \)
5 \( 1 - 1.05T + 5T^{2} \)
13 \( 1 - 3.80T + 13T^{2} \)
17 \( 1 - 2.56T + 17T^{2} \)
19 \( 1 + 0.180T + 19T^{2} \)
23 \( 1 + 0.433T + 23T^{2} \)
29 \( 1 - 9.03T + 29T^{2} \)
31 \( 1 + 0.492T + 31T^{2} \)
37 \( 1 + 0.775T + 37T^{2} \)
41 \( 1 - 3.09T + 41T^{2} \)
43 \( 1 + 1.61T + 43T^{2} \)
47 \( 1 + 0.502T + 47T^{2} \)
53 \( 1 - 9.94T + 53T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 - 5.93T + 61T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + 4.27T + 73T^{2} \)
79 \( 1 - 1.37T + 79T^{2} \)
83 \( 1 - 7.47T + 83T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−7.951666934288750887533799207075, −7.46747726494006141008100768106, −6.48353804257680522625669559950, −6.17734506355066650121435962333, −5.21022517842721892741453919909, −4.24908275847505138178801115853, −3.33598831039095057394173645111, −2.38544360101993884856538465336, −1.48779132614420047696544846698, −0.68153743558677321703945468756, 0.68153743558677321703945468756, 1.48779132614420047696544846698, 2.38544360101993884856538465336, 3.33598831039095057394173645111, 4.24908275847505138178801115853, 5.21022517842721892741453919909, 6.17734506355066650121435962333, 6.48353804257680522625669559950, 7.46747726494006141008100768106, 7.951666934288750887533799207075

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