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  + 1.80·2-s + 1.25·4-s − 2.77·5-s + 7-s − 1.33·8-s − 5.01·10-s − 2.31·13-s + 1.80·14-s − 4.93·16-s + 2.66·17-s − 8.08·19-s − 3.49·20-s + 2.43·23-s + 2.71·25-s − 4.17·26-s + 1.25·28-s − 7.55·29-s + 9.24·31-s − 6.23·32-s + 4.80·34-s − 2.77·35-s + 11.1·37-s − 14.5·38-s + 3.71·40-s − 0.299·41-s + 7.29·43-s + 4.38·46-s + ⋯
L(s)  = 1  + 1.27·2-s + 0.629·4-s − 1.24·5-s + 0.377·7-s − 0.472·8-s − 1.58·10-s − 0.641·13-s + 0.482·14-s − 1.23·16-s + 0.645·17-s − 1.85·19-s − 0.782·20-s + 0.506·23-s + 0.542·25-s − 0.819·26-s + 0.238·28-s − 1.40·29-s + 1.66·31-s − 1.10·32-s + 0.824·34-s − 0.469·35-s + 1.84·37-s − 2.36·38-s + 0.586·40-s − 0.0468·41-s + 1.11·43-s + 0.646·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.212120541$
$L(\frac12)$  $\approx$  $2.212120541$
$L(\frac{3}{2})$   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 \( 1 - 1.80T + 2T^{2} \)
5 \( 1 + 2.77T + 5T^{2} \)
13 \( 1 + 2.31T + 13T^{2} \)
17 \( 1 - 2.66T + 17T^{2} \)
19 \( 1 + 8.08T + 19T^{2} \)
23 \( 1 - 2.43T + 23T^{2} \)
29 \( 1 + 7.55T + 29T^{2} \)
31 \( 1 - 9.24T + 31T^{2} \)
37 \( 1 - 11.1T + 37T^{2} \)
41 \( 1 + 0.299T + 41T^{2} \)
43 \( 1 - 7.29T + 43T^{2} \)
47 \( 1 + 0.457T + 47T^{2} \)
53 \( 1 - 5.19T + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 - 1.81T + 61T^{2} \)
67 \( 1 + 1.42T + 67T^{2} \)
71 \( 1 + 0.558T + 71T^{2} \)
73 \( 1 - 9.66T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + 8.51T + 83T^{2} \)
89 \( 1 - 1.56T + 89T^{2} \)
97 \( 1 + 0.533T + 97T^{2} \)
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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

−7.80583616518033909102709544076, −7.12900923168911342966838071312, −6.29478016743337221925177835049, −5.70782290488733776496366539078, −4.71902378739701140950955025629, −4.39622578016523052636793643506, −3.78968112723148882206971895310, −2.94122104989068526483613529609, −2.17363011524403984393700531652, −0.58767375107835252409120330919, 0.58767375107835252409120330919, 2.17363011524403984393700531652, 2.94122104989068526483613529609, 3.78968112723148882206971895310, 4.39622578016523052636793643506, 4.71902378739701140950955025629, 5.70782290488733776496366539078, 6.29478016743337221925177835049, 7.12900923168911342966838071312, 7.80583616518033909102709544076

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