Properties

Degree $2$
Conductor $333795$
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 − 5-s − 6-s + 7-s + 3·8-s + 9-s + 10-s − 11-s − 12-s − 2·13-s − 14-s − 15-s − 16-s − 18-s − 4·19-s + 20-s + 21-s + 22-s − 8·23-s + 3·24-s + 25-s + 2·26-s + 27-s − 28-s − 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s + 0.213·22-s − 1.66·23-s + 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 333795 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333795 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(333795\)    =    \(3 \cdot 5 \cdot 7 \cdot 11 \cdot 17^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{333795} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 333795,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
17 \( 1 \)
good2 \( 1 + T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−12.84649418878023, −12.53420884517791, −11.87391073740832, −11.30026547500003, −10.96319529254328, −10.38099132139170, −10.03162538618150, −9.612345037408057, −9.009436087256759, −8.791158559120971, −8.200218736476601, −7.845147739313283, −7.474520835893896, −7.149066601822897, −6.286881361877168, −5.880427185764543, −5.074884049650902, −4.768657843066756, −4.264547402918983, −3.605607644824144, −3.432375115278704, −2.406437706094236, −1.820882537787085, −1.654667216483660, −0.4938602556940291, 0, 0.4938602556940291, 1.654667216483660, 1.820882537787085, 2.406437706094236, 3.432375115278704, 3.605607644824144, 4.264547402918983, 4.768657843066756, 5.074884049650902, 5.880427185764543, 6.286881361877168, 7.149066601822897, 7.474520835893896, 7.845147739313283, 8.200218736476601, 8.791158559120971, 9.009436087256759, 9.612345037408057, 10.03162538618150, 10.38099132139170, 10.96319529254328, 11.30026547500003, 11.87391073740832, 12.53420884517791, 12.84649418878023

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