Properties

Degree $2$
Conductor $1110$
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 + 5-s − 6-s + 8-s + 9-s + 10-s + 4·11-s − 12-s − 2·13-s − 15-s + 16-s + 2·17-s + 18-s + 4·19-s + 20-s + 4·22-s − 8·23-s − 24-s + 25-s − 2·26-s − 27-s − 2·29-s − 30-s + 8·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.852·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.371·29-s − 0.182·30-s + 1.43·31-s + 0.176·32-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: $\chi_{1110} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.442342719\)
\(L(\frac12)\) \(\approx\) \(2.442342719\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
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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

−9.860059891719032439624269133510, −9.312841926862842269895899193033, −7.964758105018483654956945009114, −7.16335427488883259355203382770, −6.17401721155628714889399945791, −5.74156590644067315350956430404, −4.62421216249512633753322007527, −3.85282303033776494959844093425, −2.54472475294312614078397255249, −1.22327620255314703849751670838, 1.22327620255314703849751670838, 2.54472475294312614078397255249, 3.85282303033776494959844093425, 4.62421216249512633753322007527, 5.74156590644067315350956430404, 6.17401721155628714889399945791, 7.16335427488883259355203382770, 7.964758105018483654956945009114, 9.312841926862842269895899193033, 9.860059891719032439624269133510

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