Properties

Label 2-2800-1.1-c1-0-51
Degree $2$
Conductor $2800$
Sign $-1$
Analytic cond. $22.3581$
Root an. cond. $4.72843$
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  + 2·3-s − 7-s + 9-s − 3·11-s − 4·13-s − 2·19-s − 2·21-s + 3·23-s − 4·27-s + 9·29-s − 8·31-s − 6·33-s + 5·37-s − 8·39-s − 6·41-s − 11·43-s − 6·47-s + 49-s + 6·53-s − 4·57-s − 10·61-s − 63-s − 5·67-s + 6·69-s − 15·71-s − 10·73-s + 3·77-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s − 0.458·19-s − 0.436·21-s + 0.625·23-s − 0.769·27-s + 1.67·29-s − 1.43·31-s − 1.04·33-s + 0.821·37-s − 1.28·39-s − 0.937·41-s − 1.67·43-s − 0.875·47-s + 1/7·49-s + 0.824·53-s − 0.529·57-s − 1.28·61-s − 0.125·63-s − 0.610·67-s + 0.722·69-s − 1.78·71-s − 1.17·73-s + 0.341·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(22.3581\)
Root analytic conductor: \(4.72843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−8.566812359090603150503102451282, −7.68315024807606221901788693933, −7.17960348065322712706483311880, −6.20114455738740303118056239220, −5.18158189203846120206831213223, −4.45747888027929058899855558527, −3.23472327777416951666174107639, −2.80135735853740060237672958376, −1.84458629259556074682826509920, 0, 1.84458629259556074682826509920, 2.80135735853740060237672958376, 3.23472327777416951666174107639, 4.45747888027929058899855558527, 5.18158189203846120206831213223, 6.20114455738740303118056239220, 7.17960348065322712706483311880, 7.68315024807606221901788693933, 8.566812359090603150503102451282

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