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  =  $(2,\ 35739,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5507939958$
$L(\frac12)$  $\approx$  $0.5507939958$
$L(\frac{3}{2})$   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

−14.89567048058011, −14.29099264812196, −13.98878385886755, −13.13689115994109, −12.91263623981106, −12.45033232198027, −11.69794757338005, −11.26400511903169, −10.25397287075130, −9.916666055701399, −9.792117682918316, −9.111946786359788, −8.640437095650848, −7.837699323574637, −7.457392986691498, −6.874063929970993, −6.156833091956870, −5.666845295987157, −4.765969755101481, −4.395245481684200, −3.387865718853382, −3.109872866915857, −2.125155770974000, −1.173504258831728, −0.3501846047434936, 0.3501846047434936, 1.173504258831728, 2.125155770974000, 3.109872866915857, 3.387865718853382, 4.395245481684200, 4.765969755101481, 5.666845295987157, 6.156833091956870, 6.874063929970993, 7.457392986691498, 7.837699323574637, 8.640437095650848, 9.111946786359788, 9.792117682918316, 9.916666055701399, 10.25397287075130, 11.26400511903169, 11.69794757338005, 12.45033232198027, 12.91263623981106, 13.13689115994109, 13.98878385886755, 14.29099264812196, 14.89567048058011

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