Properties

Degree $2$
Conductor $18496$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4·7-s + 9-s + 6·11-s − 2·13-s + 4·19-s − 8·21-s − 5·25-s + 4·27-s + 4·31-s − 12·33-s − 4·37-s + 4·39-s − 6·41-s − 8·43-s + 9·49-s + 6·53-s − 8·57-s − 4·61-s + 4·63-s − 8·67-s − 2·73-s + 10·75-s + 24·77-s − 8·79-s − 11·81-s − 6·89-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.51·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s + 0.917·19-s − 1.74·21-s − 25-s + 0.769·27-s + 0.718·31-s − 2.08·33-s − 0.657·37-s + 0.640·39-s − 0.937·41-s − 1.21·43-s + 9/7·49-s + 0.824·53-s − 1.05·57-s − 0.512·61-s + 0.503·63-s − 0.977·67-s − 0.234·73-s + 1.15·75-s + 2.73·77-s − 0.900·79-s − 1.22·81-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18496\)    =    \(2^{6} \cdot 17^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{18496} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 18496,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
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 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.29711096184687, −15.39747615168092, −14.92363914655644, −14.44708205133753, −13.84393030905019, −13.52275280778165, −12.26540000782947, −12.01139492201885, −11.66961414376930, −11.28507169785312, −10.62349920259001, −9.951413823723441, −9.394327308094559, −8.594919901466300, −8.177625583008488, −7.318466730682570, −6.827381204389536, −6.192770318735510, −5.454330905643491, −5.066680885432427, −4.371302662501670, −3.797397615696967, −2.731063397288197, −1.526971852502838, −1.298482570067965, 0, 1.298482570067965, 1.526971852502838, 2.731063397288197, 3.797397615696967, 4.371302662501670, 5.066680885432427, 5.454330905643491, 6.192770318735510, 6.827381204389536, 7.318466730682570, 8.177625583008488, 8.594919901466300, 9.394327308094559, 9.951413823723441, 10.62349920259001, 11.28507169785312, 11.66961414376930, 12.01139492201885, 12.26540000782947, 13.52275280778165, 13.84393030905019, 14.44708205133753, 14.92363914655644, 15.39747615168092, 16.29711096184687

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