Properties

Degree $2$
Conductor $303450$
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-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 12-s − 2·13-s − 14-s + 16-s − 18-s − 4·19-s + 21-s − 24-s + 2·26-s + 27-s + 28-s + 6·29-s + 4·31-s − 32-s + 36-s + 2·37-s + 4·38-s − 2·39-s − 6·41-s − 42-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.218·21-s − 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 1/6·36-s + 0.328·37-s + 0.648·38-s − 0.320·39-s − 0.937·41-s − 0.154·42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(303450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{303450} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
17 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
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 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−12.81246910342888, −12.33636554742384, −12.01533055178003, −11.55277423737638, −10.88450174223793, −10.56056310922334, −10.12066613282167, −9.715982485348107, −9.111440824298888, −8.782812049249457, −8.296140685821774, −7.908085136651757, −7.480389118127940, −6.953244992297781, −6.329420082730196, −6.170332924337920, −5.230771829946871, −4.750552607338308, −4.392403257379195, −3.605077939085955, −3.123957150500381, −2.482546887782698, −2.094084121745072, −1.453125232623760, −0.7981162138534166, 0, 0.7981162138534166, 1.453125232623760, 2.094084121745072, 2.482546887782698, 3.123957150500381, 3.605077939085955, 4.392403257379195, 4.750552607338308, 5.230771829946871, 6.170332924337920, 6.329420082730196, 6.953244992297781, 7.480389118127940, 7.908085136651757, 8.296140685821774, 8.782812049249457, 9.111440824298888, 9.715982485348107, 10.12066613282167, 10.56056310922334, 10.88450174223793, 11.55277423737638, 12.01533055178003, 12.33636554742384, 12.81246910342888

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