Properties

Label 2-2240-1.1-c1-0-20
Degree $2$
Conductor $2240$
Sign $1$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s − 3·9-s + 4·11-s + 6·13-s + 2·17-s + 25-s − 6·29-s − 8·31-s + 35-s + 10·37-s + 2·41-s + 4·43-s − 3·45-s − 8·47-s + 49-s + 2·53-s + 4·55-s − 8·59-s + 14·61-s − 3·63-s + 6·65-s − 12·67-s + 16·71-s + 2·73-s + 4·77-s + 8·79-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 9-s + 1.20·11-s + 1.66·13-s + 0.485·17-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.169·35-s + 1.64·37-s + 0.312·41-s + 0.609·43-s − 0.447·45-s − 1.16·47-s + 1/7·49-s + 0.274·53-s + 0.539·55-s − 1.04·59-s + 1.79·61-s − 0.377·63-s + 0.744·65-s − 1.46·67-s + 1.89·71-s + 0.234·73-s + 0.455·77-s + 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2240} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.216596921\)
\(L(\frac12)\) \(\approx\) \(2.216596921\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
good3 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
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

−9.118400765890595835755961485120, −8.352657010340947474880201780838, −7.60394710404495150534961844752, −6.45059958255616921205327535779, −5.97923649902359659099455023124, −5.24128565523298539791335980836, −3.98731345269432541606781465630, −3.37733882756944247929081378523, −2.06009477379491109088643157397, −1.04018234489920160371252286291, 1.04018234489920160371252286291, 2.06009477379491109088643157397, 3.37733882756944247929081378523, 3.98731345269432541606781465630, 5.24128565523298539791335980836, 5.97923649902359659099455023124, 6.45059958255616921205327535779, 7.60394710404495150534961844752, 8.352657010340947474880201780838, 9.118400765890595835755961485120

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