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  − 2·4-s + 3·5-s − 4·7-s − 11-s − 2·13-s + 4·16-s − 6·20-s − 3·23-s + 4·25-s + 8·28-s − 6·29-s + 7·31-s − 12·35-s + 7·37-s − 10·43-s + 2·44-s + 9·49-s + 4·52-s + 6·53-s − 3·55-s + 3·59-s − 10·61-s − 8·64-s − 6·65-s − 11·67-s + 15·71-s + 8·73-s + ⋯
L(s)  = 1  − 4-s + 1.34·5-s − 1.51·7-s − 0.301·11-s − 0.554·13-s + 16-s − 1.34·20-s − 0.625·23-s + 4/5·25-s + 1.51·28-s − 1.11·29-s + 1.25·31-s − 2.02·35-s + 1.15·37-s − 1.52·43-s + 0.301·44-s + 9/7·49-s + 0.554·52-s + 0.824·53-s − 0.404·55-s + 0.390·59-s − 1.28·61-s − 64-s − 0.744·65-s − 1.34·67-s + 1.78·71-s + 0.936·73-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 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
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

−15.06708946880622, −14.63668707001443, −13.83512556844676, −13.57839350482898, −13.24328232292401, −12.73598295550968, −12.25889222528966, −11.63854086723173, −10.60401783478078, −10.20678141660752, −9.814704066970488, −9.293979404701926, −9.177049514743532, −8.177249587662250, −7.751452458000772, −6.807207165204294, −6.339018606507991, −5.850378673541287, −5.292844108859432, −4.703719604085691, −3.899712242735085, −3.280175421382055, −2.595841183025195, −1.919968475270899, −0.8434065818445668, 0, 0.8434065818445668, 1.919968475270899, 2.595841183025195, 3.280175421382055, 3.899712242735085, 4.703719604085691, 5.292844108859432, 5.850378673541287, 6.339018606507991, 6.807207165204294, 7.751452458000772, 8.177249587662250, 9.177049514743532, 9.293979404701926, 9.814704066970488, 10.20678141660752, 10.60401783478078, 11.63854086723173, 12.25889222528966, 12.73598295550968, 13.24328232292401, 13.57839350482898, 13.83512556844676, 14.63668707001443, 15.06708946880622

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