Properties

Degree 2
Conductor $ 2^{4} \cdot 7 \cdot 13^{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  + 2·3-s + 7-s + 9-s + 6·17-s + 2·19-s + 2·21-s − 5·25-s − 4·27-s − 6·29-s − 4·31-s − 2·37-s − 6·41-s − 8·43-s − 12·47-s + 49-s + 12·51-s + 6·53-s + 4·57-s − 6·59-s + 8·61-s + 63-s − 4·67-s − 2·73-s − 10·75-s − 8·79-s − 11·81-s − 6·83-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.45·17-s + 0.458·19-s + 0.436·21-s − 25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s − 0.328·37-s − 0.937·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s + 1.68·51-s + 0.824·53-s + 0.529·57-s − 0.781·59-s + 1.02·61-s + 0.125·63-s − 0.488·67-s − 0.234·73-s − 1.15·75-s − 0.900·79-s − 1.22·81-s − 0.658·83-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 18928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 18928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(18928\)    =    \(2^{4} \cdot 7 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{18928} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 18928,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
13 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
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

−15.90460387492932, −15.28202131409128, −14.77894655183596, −14.43732633925939, −13.94989067595884, −13.27859479228289, −13.00617589634972, −12.03308542385990, −11.68485944588376, −11.11500962821117, −10.14507580780563, −9.881054731467372, −9.233249826071295, −8.620286924268313, −8.088084102246574, −7.614113324832928, −7.135082173089946, −6.172186020675195, −5.463874135574944, −5.013011607079751, −3.895326142717217, −3.507124542133636, −2.898275228820932, −1.935318101691155, −1.444995703165364, 0, 1.444995703165364, 1.935318101691155, 2.898275228820932, 3.507124542133636, 3.895326142717217, 5.013011607079751, 5.463874135574944, 6.172186020675195, 7.135082173089946, 7.614113324832928, 8.088084102246574, 8.620286924268313, 9.233249826071295, 9.881054731467372, 10.14507580780563, 11.11500962821117, 11.68485944588376, 12.03308542385990, 13.00617589634972, 13.27859479228289, 13.94989067595884, 14.43732633925939, 14.77894655183596, 15.28202131409128, 15.90460387492932

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