Properties

Degree 2
Conductor $ 2 \cdot 7 \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 + 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 + 2·21-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 + 8·39-s − 6·41-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 + 0.436·21-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 + 1.28·39-s − 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5054\)    =    \(2 \cdot 7 \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{5054} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 5054,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(5.473593818\)
\(L(\frac12)\)  \(\approx\)  \(5.473593818\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 - T \)
19 \( 1 \)
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 \( 1 - 6 T + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
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

−17.58683895612338, −17.20428237109858, −16.04443278740800, −15.96521631721081, −15.10755471928217, −14.50158664857420, −14.14516671250556, −13.58715883443369, −13.08589258268545, −12.23650667249491, −11.66721945283422, −11.04500337443535, −10.12244142157992, −9.677190571290227, −8.651374782407252, −8.196197307405668, −7.722005162091900, −6.772980055363700, −5.979070887547406, −5.311554224321026, −4.359061789115610, −3.552460930723167, −3.097386428585368, −2.114582720208594, −1.194601328137030, 1.194601328137030, 2.114582720208594, 3.097386428585368, 3.552460930723167, 4.359061789115610, 5.311554224321026, 5.979070887547406, 6.772980055363700, 7.722005162091900, 8.196197307405668, 8.651374782407252, 9.677190571290227, 10.12244142157992, 11.04500337443535, 11.66721945283422, 12.23650667249491, 13.08589258268545, 13.58715883443369, 14.14516671250556, 14.50158664857420, 15.10755471928217, 15.96521631721081, 16.04443278740800, 17.20428237109858, 17.58683895612338

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