Properties

Label 2-2800-1.1-c1-0-16
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s − 2·9-s + 11-s + 13-s + 3·17-s + 4·19-s − 21-s + 2·23-s − 5·27-s − 29-s + 6·31-s + 33-s − 2·37-s + 39-s − 10·41-s + 9·47-s + 49-s + 3·51-s + 14·53-s + 4·57-s − 6·59-s − 4·61-s + 2·63-s + 10·67-s + 2·69-s + 16·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.301·11-s + 0.277·13-s + 0.727·17-s + 0.917·19-s − 0.218·21-s + 0.417·23-s − 0.962·27-s − 0.185·29-s + 1.07·31-s + 0.174·33-s − 0.328·37-s + 0.160·39-s − 1.56·41-s + 1.31·47-s + 1/7·49-s + 0.420·51-s + 1.92·53-s + 0.529·57-s − 0.781·59-s − 0.512·61-s + 0.251·63-s + 1.22·67-s + 0.240·69-s + 1.89·71-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: \(0\)
Selberg data: \((2,\ 2800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.134540700\)
\(L(\frac12)\) \(\approx\) \(2.134540700\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−8.814515947158325712627809437328, −8.123666350703995509415125105866, −7.38611806680500244103731048032, −6.54711154296763061960540460563, −5.71515738612701129421208895231, −4.99175613309792938801335562022, −3.73356781126855872540408537672, −3.21242139256750036523526171827, −2.25066718620878905986267476540, −0.892435010480354783030345469894, 0.892435010480354783030345469894, 2.25066718620878905986267476540, 3.21242139256750036523526171827, 3.73356781126855872540408537672, 4.99175613309792938801335562022, 5.71515738612701129421208895231, 6.54711154296763061960540460563, 7.38611806680500244103731048032, 8.123666350703995509415125105866, 8.814515947158325712627809437328

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