Properties

Degree $2$
Conductor $1110$
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 − 4·7-s − 8-s + 9-s + 10-s + 4·11-s − 12-s + 2·13-s + 4·14-s + 15-s + 16-s − 2·17-s − 18-s + 8·19-s − 20-s + 4·21-s − 4·22-s + 24-s + 25-s − 2·26-s − 27-s − 4·28-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 − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.554·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 1.83·19-s − 0.223·20-s + 0.872·21-s − 0.852·22-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.755·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{1110} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
37 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−19.94446274253394, −19.73314027311207, −18.75479320812171, −18.41460904583648, −17.51409053995308, −16.75768072247060, −16.20475116312069, −15.84623980443286, −14.96640541928155, −13.97749055769794, −13.05451746557234, −12.43159026281796, −11.52887185980547, −11.18222396378824, −10.04105229915318, −9.435621590245950, −8.909761485587000, −7.639715523538763, −6.891214434606250, −6.307205494307446, −5.368070285372935, −3.848181817948898, −3.225410955332902, −1.429541736276097, 0, 1.429541736276097, 3.225410955332902, 3.848181817948898, 5.368070285372935, 6.307205494307446, 6.891214434606250, 7.639715523538763, 8.909761485587000, 9.435621590245950, 10.04105229915318, 11.18222396378824, 11.52887185980547, 12.43159026281796, 13.05451746557234, 13.97749055769794, 14.96640541928155, 15.84623980443286, 16.20475116312069, 16.75768072247060, 17.51409053995308, 18.41460904583648, 18.75479320812171, 19.73314027311207, 19.94446274253394

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