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 − 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  =  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 \( 1 + p T^{2} \)
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

−15.43649008822520, −14.66378545430936, −13.88132733832406, −13.57866606547949, −13.18007344408834, −12.62495612142740, −12.21202953927662, −11.19325278458936, −10.87328307791663, −10.35649134311480, −9.672393437734619, −9.249175120066005, −9.045116397963294, −8.146728226537042, −7.841339497555713, −7.021051172195012, −6.544004517140107, −5.819675896683907, −5.460013222250428, −4.376486498046892, −3.876946640051596, −3.465807814343749, −2.439188922426091, −1.759505204923776, −0.6823693033660983, 0, 0.6823693033660983, 1.759505204923776, 2.439188922426091, 3.465807814343749, 3.876946640051596, 4.376486498046892, 5.460013222250428, 5.819675896683907, 6.544004517140107, 7.021051172195012, 7.841339497555713, 8.146728226537042, 9.045116397963294, 9.249175120066005, 9.672393437734619, 10.35649134311480, 10.87328307791663, 11.19325278458936, 12.21202953927662, 12.62495612142740, 13.18007344408834, 13.57866606547949, 13.88132733832406, 14.66378545430936, 15.43649008822520

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