Properties

Label 2-2800-1.1-c1-0-25
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  − 3·3-s − 7-s + 6·9-s − 3·11-s + 13-s − 5·17-s + 8·19-s + 3·21-s − 2·23-s − 9·27-s − 29-s + 2·31-s + 9·33-s + 10·37-s − 3·39-s − 6·41-s + 4·43-s − 11·47-s + 49-s + 15·51-s + 6·53-s − 24·57-s + 10·59-s − 6·63-s + 10·67-s + 6·69-s − 10·73-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.377·7-s + 2·9-s − 0.904·11-s + 0.277·13-s − 1.21·17-s + 1.83·19-s + 0.654·21-s − 0.417·23-s − 1.73·27-s − 0.185·29-s + 0.359·31-s + 1.56·33-s + 1.64·37-s − 0.480·39-s − 0.937·41-s + 0.609·43-s − 1.60·47-s + 1/7·49-s + 2.10·51-s + 0.824·53-s − 3.17·57-s + 1.30·59-s − 0.755·63-s + 1.22·67-s + 0.722·69-s − 1.17·73-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 \( 1 + p T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−8.310462843131967526259094880023, −7.43604934325170529521436248326, −6.74624844834157900794873610963, −6.05756152589644193767265190926, −5.36218497973593606756832966709, −4.78170159849224593763535490779, −3.80716066528279903975558995395, −2.55566948558996585665561420469, −1.12599312477747894645817160049, 0, 1.12599312477747894645817160049, 2.55566948558996585665561420469, 3.80716066528279903975558995395, 4.78170159849224593763535490779, 5.36218497973593606756832966709, 6.05756152589644193767265190926, 6.74624844834157900794873610963, 7.43604934325170529521436248326, 8.310462843131967526259094880023

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