Properties

Degree $2$
Conductor $4410$
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 + 4-s − 5-s − 8-s + 10-s − 4·11-s + 6·13-s + 16-s + 2·17-s − 20-s + 4·22-s + 25-s − 6·26-s − 6·29-s − 8·31-s − 32-s − 2·34-s − 10·37-s + 40-s + 2·41-s + 4·43-s − 4·44-s + 8·47-s − 50-s + 6·52-s + 2·53-s + 4·55-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.20·11-s + 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.223·20-s + 0.852·22-s + 1/5·25-s − 1.17·26-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s − 1.64·37-s + 0.158·40-s + 0.312·41-s + 0.609·43-s − 0.603·44-s + 1.16·47-s − 0.141·50-s + 0.832·52-s + 0.274·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{4410} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−18.28940623462290, −17.82821688707575, −17.02437045320870, −16.32205933834239, −15.92150168817526, −15.43931703025608, −14.75790621544189, −13.91667883892001, −13.25981030547693, −12.65423344024963, −11.97481353581502, −11.03690252556506, −10.84999530418765, −10.18253374073535, −9.187318786676869, −8.751814240447310, −7.957237096013159, −7.507930505304021, −6.716545079839880, −5.734681629344815, −5.299920328479471, −3.944491752690073, −3.399151866107796, −2.306092052395085, −1.256351411655489, 0, 1.256351411655489, 2.306092052395085, 3.399151866107796, 3.944491752690073, 5.299920328479471, 5.734681629344815, 6.716545079839880, 7.507930505304021, 7.957237096013159, 8.751814240447310, 9.187318786676869, 10.18253374073535, 10.84999530418765, 11.03690252556506, 11.97481353581502, 12.65423344024963, 13.25981030547693, 13.91667883892001, 14.75790621544189, 15.43931703025608, 15.92150168817526, 16.32205933834239, 17.02437045320870, 17.82821688707575, 18.28940623462290

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