Properties

Label 2-2800-1.1-c1-0-49
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  + 3-s + 7-s − 2·9-s − 3·11-s + 13-s + 3·17-s − 2·19-s + 21-s − 6·23-s − 5·27-s − 9·29-s − 8·31-s − 3·33-s + 10·37-s + 39-s + 2·43-s − 3·47-s + 49-s + 3·51-s − 2·57-s − 12·59-s + 8·61-s − 2·63-s + 8·67-s − 6·69-s − 14·73-s − 3·77-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.277·13-s + 0.727·17-s − 0.458·19-s + 0.218·21-s − 1.25·23-s − 0.962·27-s − 1.67·29-s − 1.43·31-s − 0.522·33-s + 1.64·37-s + 0.160·39-s + 0.304·43-s − 0.437·47-s + 1/7·49-s + 0.420·51-s − 0.264·57-s − 1.56·59-s + 1.02·61-s − 0.251·63-s + 0.977·67-s − 0.722·69-s − 1.63·73-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: $\chi_{2800} (1, \cdot )$
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 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−8.159719225797647385384348981545, −7.970794681732430004914952341948, −7.12249544742866899506814825905, −5.77447259069469215023304609016, −5.62167942609611010380190274624, −4.34415209537200474255903721498, −3.53342468758319092716695340957, −2.60469033876025189928088675517, −1.74379530940863800738439159468, 0, 1.74379530940863800738439159468, 2.60469033876025189928088675517, 3.53342468758319092716695340957, 4.34415209537200474255903721498, 5.62167942609611010380190274624, 5.77447259069469215023304609016, 7.12249544742866899506814825905, 7.970794681732430004914952341948, 8.159719225797647385384348981545

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