Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13 $
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 + 4-s + 5-s − 8-s − 10-s − 13-s + 16-s + 6·17-s + 4·19-s + 20-s + 25-s + 26-s − 6·29-s + 4·31-s − 32-s − 6·34-s − 10·37-s − 4·38-s − 40-s + 6·41-s + 8·43-s − 50-s − 52-s + 6·53-s + 6·58-s − 12·59-s − 14·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.223·20-s + 1/5·25-s + 0.196·26-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s − 1.64·37-s − 0.648·38-s − 0.158·40-s + 0.937·41-s + 1.21·43-s − 0.141·50-s − 0.138·52-s + 0.824·53-s + 0.787·58-s − 1.56·59-s − 1.79·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(57330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{57330} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 57330,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.884176872$
$L(\frac12)$  $\approx$  $1.884176872$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 + T \)
good11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
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 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
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 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.32348181170331, −13.92350109334275, −13.47849538724694, −12.62767356785222, −12.30647320525934, −11.85260741399686, −11.23892747554515, −10.59773187345517, −10.26742879586816, −9.684631745651272, −9.225689656298729, −8.856522280236919, −8.031599481029687, −7.498772768335837, −7.332232828071617, −6.449873957000038, −5.847953598509866, −5.480573564458334, −4.814023889428378, −3.984432514400027, −3.216689469550975, −2.814891439965096, −1.893257224341741, −1.332612693755464, −0.5548873948317094, 0.5548873948317094, 1.332612693755464, 1.893257224341741, 2.814891439965096, 3.216689469550975, 3.984432514400027, 4.814023889428378, 5.480573564458334, 5.847953598509866, 6.449873957000038, 7.332232828071617, 7.498772768335837, 8.031599481029687, 8.856522280236919, 9.225689656298729, 9.684631745651272, 10.26742879586816, 10.59773187345517, 11.23892747554515, 11.85260741399686, 12.30647320525934, 12.62767356785222, 13.47849538724694, 13.92350109334275, 14.32348181170331

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