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-s − 4-s + 2·5-s + 7-s − 3·8-s + 2·10-s − 4·13-s + 14-s − 16-s + 4·17-s − 2·20-s + 4·23-s − 25-s − 4·26-s − 28-s − 6·29-s + 10·31-s + 5·32-s + 4·34-s + 2·35-s − 6·37-s − 6·40-s + 4·41-s − 12·43-s + 4·46-s + 10·47-s + 49-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s − 1.06·8-s + 0.632·10-s − 1.10·13-s + 0.267·14-s − 1/4·16-s + 0.970·17-s − 0.447·20-s + 0.834·23-s − 1/5·25-s − 0.784·26-s − 0.188·28-s − 1.11·29-s + 1.79·31-s + 0.883·32-s + 0.685·34-s + 0.338·35-s − 0.986·37-s − 0.948·40-s + 0.624·41-s − 1.82·43-s + 0.589·46-s + 1.45·47-s + 1/7·49-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  :  $\chi_{7623} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.769521394$
$L(\frac12)$  $\approx$  $2.769521394$
$L(\frac{3}{2})$   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 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−17.07844899689479, −16.80015186954418, −15.55012501408589, −15.21930841532192, −14.47679875548270, −14.14888268129216, −13.55702999164986, −13.09095790457201, −12.30632173292382, −12.00865906178513, −11.20874838230503, −10.23997905567830, −9.852903890863240, −9.300515951584338, −8.576763002125085, −7.877394684428562, −7.070070003452164, −6.312515429823247, −5.452178119776358, −5.228945114316478, −4.457722518217533, −3.589224714932761, −2.783498161575404, −1.961678779181671, −0.7500680530818427, 0.7500680530818427, 1.961678779181671, 2.783498161575404, 3.589224714932761, 4.457722518217533, 5.228945114316478, 5.452178119776358, 6.312515429823247, 7.070070003452164, 7.877394684428562, 8.576763002125085, 9.300515951584338, 9.852903890863240, 10.23997905567830, 11.20874838230503, 12.00865906178513, 12.30632173292382, 13.09095790457201, 13.55702999164986, 14.14888268129216, 14.47679875548270, 15.21930841532192, 15.55012501408589, 16.80015186954418, 17.07844899689479

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