Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 11 \cdot 19^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s − 4-s − 2·5-s + 6-s + 7-s − 3·8-s + 9-s − 2·10-s − 11-s − 12-s − 6·13-s + 14-s − 2·15-s − 16-s + 2·17-s + 18-s + 2·20-s + 21-s − 22-s − 3·24-s − 25-s − 6·26-s + 27-s − 28-s + 2·29-s − 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.447·20-s + 0.218·21-s − 0.213·22-s − 0.612·24-s − 1/5·25-s − 1.17·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(83391\)    =    \(3 \cdot 7 \cdot 11 \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{83391} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 83391,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
19 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.37057914884708, −14.06366267630270, −13.55519785747596, −12.87643485468884, −12.51890455191232, −12.15065152982820, −11.64779985043995, −11.16025405737205, −10.36058630794687, −9.835334615784587, −9.548872396507056, −8.768103014782007, −8.376942332333106, −7.882044659935690, −7.366943205074850, −6.942193908180665, −6.159177601176456, −5.316150615592865, −5.026444844056619, −4.574011503419736, −3.904491351667717, −3.373726018039744, −2.972048188477216, −2.151083353293982, −1.432398174032259, 0, 0, 1.432398174032259, 2.151083353293982, 2.972048188477216, 3.373726018039744, 3.904491351667717, 4.574011503419736, 5.026444844056619, 5.316150615592865, 6.159177601176456, 6.942193908180665, 7.366943205074850, 7.882044659935690, 8.376942332333106, 8.768103014782007, 9.548872396507056, 9.835334615784587, 10.36058630794687, 11.16025405737205, 11.64779985043995, 12.15065152982820, 12.51890455191232, 12.87643485468884, 13.55519785747596, 14.06366267630270, 14.37057914884708

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