Properties

Degree $2$
Conductor $86394$
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 − 3-s + 4-s − 2·5-s + 6-s − 7-s − 8-s + 9-s + 2·10-s − 12-s + 2·13-s + 14-s + 2·15-s + 16-s − 17-s − 18-s − 4·19-s − 2·20-s + 21-s + 8·23-s + 24-s − 25-s − 2·26-s − 27-s − 28-s − 6·29-s − 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.218·21-s + 1.66·23-s + 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(86394\)    =    \(2 \cdot 3 \cdot 7 \cdot 11^{2} \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{86394} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 86394,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3537110316\)
\(L(\frac12)\) \(\approx\) \(0.3537110316\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.71221429240383, −13.38776619806918, −12.71254848960348, −12.44521419451862, −11.74454480182789, −11.32912287608752, −11.05012021434323, −10.37378478648385, −10.15188547082213, −9.231335226141974, −8.886666396559704, −8.529119655951821, −7.705040605706059, −7.360632178490935, −6.902441502838439, −6.208047249530819, −5.922369511902495, −5.010778408752697, −4.608555607460509, −3.704299505493062, −3.485102779312385, −2.604020238185382, −1.817708392081760, −1.098683894414806, −0.2507893762815717, 0.2507893762815717, 1.098683894414806, 1.817708392081760, 2.604020238185382, 3.485102779312385, 3.704299505493062, 4.608555607460509, 5.010778408752697, 5.922369511902495, 6.208047249530819, 6.902441502838439, 7.360632178490935, 7.705040605706059, 8.529119655951821, 8.886666396559704, 9.231335226141974, 10.15188547082213, 10.37378478648385, 11.05012021434323, 11.32912287608752, 11.74454480182789, 12.44521419451862, 12.71254848960348, 13.38776619806918, 13.71221429240383

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