Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 47^{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 − 2·3-s + 4-s + 2·6-s + 7-s − 8-s + 9-s − 2·12-s + 4·13-s − 14-s + 16-s + 6·17-s − 18-s − 2·19-s − 2·21-s + 2·24-s − 5·25-s − 4·26-s + 4·27-s + 28-s + 6·29-s + 4·31-s − 32-s − 6·34-s + 36-s + 2·37-s + 2·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.577·12-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.436·21-s + 0.408·24-s − 25-s − 0.784·26-s + 0.769·27-s + 0.188·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.328·37-s + 0.324·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30926 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30926 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(30926\)    =    \(2 \cdot 7 \cdot 47^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{30926} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 30926,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;47\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;47\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 - T \)
47 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.49860040194924, −15.02538542340238, −14.32174888009762, −13.76620003730495, −13.25142335299183, −12.32329851420824, −12.11479124157842, −11.49302650001710, −11.20090004313582, −10.45233746418812, −10.17383644872705, −9.616080410246592, −8.725642489789581, −8.239009925136723, −7.935307552601874, −6.976023677008769, −6.525526593178665, −5.950709823788886, −5.472310647259829, −4.870418420918839, −4.022319660034148, −3.325208938740451, −2.494655821274781, −1.440823148497990, −0.9933664688767733, 0, 0.9933664688767733, 1.440823148497990, 2.494655821274781, 3.325208938740451, 4.022319660034148, 4.870418420918839, 5.472310647259829, 5.950709823788886, 6.525526593178665, 6.976023677008769, 7.935307552601874, 8.239009925136723, 8.725642489789581, 9.616080410246592, 10.17383644872705, 10.45233746418812, 11.20090004313582, 11.49302650001710, 12.11479124157842, 12.32329851420824, 13.25142335299183, 13.76620003730495, 14.32174888009762, 15.02538542340238, 15.49860040194924

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