Properties

Label 2-2240-1.1-c1-0-39
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 7-s − 2·9-s + 3·11-s − 5·13-s − 15-s + 3·17-s − 2·19-s − 21-s − 6·23-s + 25-s + 5·27-s − 3·29-s − 4·31-s − 3·33-s + 35-s − 2·37-s + 5·39-s − 12·41-s + 10·43-s − 2·45-s + 9·47-s + 49-s − 3·51-s − 12·53-s + 3·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s − 1.38·13-s − 0.258·15-s + 0.727·17-s − 0.458·19-s − 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s − 0.718·31-s − 0.522·33-s + 0.169·35-s − 0.328·37-s + 0.800·39-s − 1.87·41-s + 1.52·43-s − 0.298·45-s + 1.31·47-s + 1/7·49-s − 0.420·51-s − 1.64·53-s + 0.404·55-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
good3 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−8.722556678528362158075028214833, −7.82866251649990434300481776759, −7.05288221116077551560339282808, −6.15246422175821288222036469083, −5.54934319897223270256186001671, −4.78822533627578751004226271654, −3.79583222083421885356258159776, −2.59983312100744179860207048134, −1.58268461056007998080671173620, 0, 1.58268461056007998080671173620, 2.59983312100744179860207048134, 3.79583222083421885356258159776, 4.78822533627578751004226271654, 5.54934319897223270256186001671, 6.15246422175821288222036469083, 7.05288221116077551560339282808, 7.82866251649990434300481776759, 8.722556678528362158075028214833

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