Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 5-s − 3·7-s − 3·8-s − 10-s − 11-s − 6·13-s − 3·14-s − 16-s + 20-s − 22-s + 2·23-s − 4·25-s − 6·26-s + 3·28-s − 4·29-s + 4·31-s + 5·32-s + 3·35-s + 7·37-s + 3·40-s + 4·41-s − 4·43-s + 44-s + 2·46-s − 6·47-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.13·7-s − 1.06·8-s − 0.316·10-s − 0.301·11-s − 1.66·13-s − 0.801·14-s − 1/4·16-s + 0.223·20-s − 0.213·22-s + 0.417·23-s − 4/5·25-s − 1.17·26-s + 0.566·28-s − 0.742·29-s + 0.718·31-s + 0.883·32-s + 0.507·35-s + 1.15·37-s + 0.474·40-s + 0.624·41-s − 0.609·43-s + 0.150·44-s + 0.294·46-s − 0.875·47-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  =  1
Selberg data  =  $(2,\ 35739,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−14.90708599567446, −14.81167371513077, −14.17526291059148, −13.35874538610824, −13.17798194912992, −12.69730977322388, −12.09824211468108, −11.78238391633125, −11.11366736127498, −10.22656657518247, −9.784048564579062, −9.453211237278435, −8.895553606586052, −7.994522846418124, −7.688154121417032, −6.919087030314725, −6.325553248793142, −5.811960446192141, −4.977705822545195, −4.742418561361602, −3.903338721174893, −3.401886584409902, −2.776526369005608, −2.154347931359361, −0.6728109310119284, 0, 0.6728109310119284, 2.154347931359361, 2.776526369005608, 3.401886584409902, 3.903338721174893, 4.742418561361602, 4.977705822545195, 5.811960446192141, 6.325553248793142, 6.919087030314725, 7.688154121417032, 7.994522846418124, 8.895553606586052, 9.453211237278435, 9.784048564579062, 10.22656657518247, 11.11366736127498, 11.78238391633125, 12.09824211468108, 12.69730977322388, 13.17798194912992, 13.35874538610824, 14.17526291059148, 14.81167371513077, 14.90708599567446

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