Properties

Degree 2
Conductor $ 2 \cdot 17 \cdot 43^{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 + 4·7-s − 8-s + 9-s + 6·11-s + 2·12-s + 2·13-s − 4·14-s + 16-s − 17-s − 18-s + 4·19-s + 8·21-s − 6·22-s − 2·24-s − 5·25-s − 2·26-s − 4·27-s + 4·28-s − 4·31-s − 32-s + 12·33-s + 34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.577·12-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s + 1.74·21-s − 1.27·22-s − 0.408·24-s − 25-s − 0.392·26-s − 0.769·27-s + 0.755·28-s − 0.718·31-s − 0.176·32-s + 2.08·33-s + 0.171·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62866 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62866 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(62866\)    =    \(2 \cdot 17 \cdot 43^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{62866} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 62866,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(4.548544497\)
\(L(\frac12)\)  \(\approx\)  \(4.548544497\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;17,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
17 \( 1 + T \)
43 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.38099055872564, −13.95153879813623, −13.49610086513020, −12.77015526921224, −11.99065817549700, −11.60018832379888, −11.26677615198893, −10.82403690478986, −9.946331754779960, −9.409904639396544, −9.085546152463763, −8.697543948492074, −7.954682111809038, −7.864679825114635, −7.215651319664046, −6.520068432191183, −5.927298940517305, −5.284078204759776, −4.452148873979697, −3.857193642760091, −3.477507202233385, −2.565091099784354, −1.896430621063719, −1.483891290427476, −0.7941619781895600, 0.7941619781895600, 1.483891290427476, 1.896430621063719, 2.565091099784354, 3.477507202233385, 3.857193642760091, 4.452148873979697, 5.284078204759776, 5.927298940517305, 6.520068432191183, 7.215651319664046, 7.864679825114635, 7.954682111809038, 8.697543948492074, 9.085546152463763, 9.409904639396544, 9.946331754779960, 10.82403690478986, 11.26677615198893, 11.60018832379888, 11.99065817549700, 12.77015526921224, 13.49610086513020, 13.95153879813623, 14.38099055872564

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