Properties

Degree 2
Conductor $ 3^{2} \cdot 11 \cdot 19^{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 − 4·7-s − 3·8-s + 11-s + 4·13-s − 4·14-s − 16-s + 4·17-s + 22-s + 4·23-s − 5·25-s + 4·26-s + 4·28-s + 2·29-s − 8·31-s + 5·32-s + 4·34-s − 6·41-s − 44-s + 4·46-s − 12·47-s + 9·49-s − 5·50-s − 4·52-s + 10·53-s + 12·56-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.51·7-s − 1.06·8-s + 0.301·11-s + 1.10·13-s − 1.06·14-s − 1/4·16-s + 0.970·17-s + 0.213·22-s + 0.834·23-s − 25-s + 0.784·26-s + 0.755·28-s + 0.371·29-s − 1.43·31-s + 0.883·32-s + 0.685·34-s − 0.937·41-s − 0.150·44-s + 0.589·46-s − 1.75·47-s + 9/7·49-s − 0.707·50-s − 0.554·52-s + 1.37·53-s + 1.60·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(35739\)    =    \(3^{2} \cdot 11 \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{35739} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 35739,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.660876800$
$L(\frac12)$  $\approx$  $1.660876800$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−14.80313276962203, −14.37732366368343, −13.70200502702764, −13.28075088431346, −13.05588554337162, −12.35267157906488, −12.05770205421925, −11.30933003020916, −10.73767216605605, −9.960533834409158, −9.547914794786065, −9.237492980832721, −8.430412111035658, −8.076065646360248, −7.033720315618199, −6.595570892041288, −6.108515712153598, −5.402522576222803, −5.129438811057500, −3.910416818448163, −3.725239182532779, −3.278055053911491, −2.483617801830810, −1.346000477884718, −0.4534245420940150, 0.4534245420940150, 1.346000477884718, 2.483617801830810, 3.278055053911491, 3.725239182532779, 3.910416818448163, 5.129438811057500, 5.402522576222803, 6.108515712153598, 6.595570892041288, 7.033720315618199, 8.076065646360248, 8.430412111035658, 9.237492980832721, 9.547914794786065, 9.960533834409158, 10.73767216605605, 11.30933003020916, 12.05770205421925, 12.35267157906488, 13.05588554337162, 13.28075088431346, 13.70200502702764, 14.37732366368343, 14.80313276962203

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