Properties

Degree $2$
Conductor $53312$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 5-s + 9-s − 4·11-s − 2·13-s − 2·15-s − 17-s − 19-s + 23-s − 4·25-s + 4·27-s + 6·29-s + 4·31-s + 8·33-s + 3·37-s + 4·39-s − 10·41-s + 43-s + 45-s + 4·47-s + 2·51-s − 6·53-s − 4·55-s + 2·57-s − 59-s + 10·61-s − 2·65-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s − 0.242·17-s − 0.229·19-s + 0.208·23-s − 4/5·25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s + 1.39·33-s + 0.493·37-s + 0.640·39-s − 1.56·41-s + 0.152·43-s + 0.149·45-s + 0.583·47-s + 0.280·51-s − 0.824·53-s − 0.539·55-s + 0.264·57-s − 0.130·59-s + 1.28·61-s − 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53312\)    =    \(2^{6} \cdot 7^{2} \cdot 17\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{53312} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 53312,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
17 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.75069565024614, −14.03276448985293, −13.70985382675716, −13.02196679100362, −12.71026078593208, −11.99813532635673, −11.73164521107210, −11.09185732907609, −10.61063061266641, −10.03231735959499, −9.920687876751364, −8.973304775144183, −8.420179954633661, −7.865162797267148, −7.274733323813025, −6.486214539149495, −6.307097204756906, −5.486506945645918, −5.156980707773537, −4.684827733206032, −3.957059168441883, −2.923634184996439, −2.532868362122679, −1.692006856643341, −0.7263311737616488, 0, 0.7263311737616488, 1.692006856643341, 2.532868362122679, 2.923634184996439, 3.957059168441883, 4.684827733206032, 5.156980707773537, 5.486506945645918, 6.307097204756906, 6.486214539149495, 7.274733323813025, 7.865162797267148, 8.420179954633661, 8.973304775144183, 9.920687876751364, 10.03231735959499, 10.61063061266641, 11.09185732907609, 11.73164521107210, 11.99813532635673, 12.71026078593208, 13.02196679100362, 13.70985382675716, 14.03276448985293, 14.75069565024614

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