Properties

Label 2-2800-1.1-c1-0-7
Degree $2$
Conductor $2800$
Sign $1$
Analytic cond. $22.3581$
Root an. cond. $4.72843$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.23·3-s + 7-s + 7.47·9-s + 0.236·11-s − 1.23·13-s − 2.47·17-s + 4.47·19-s − 3.23·21-s + 6.23·23-s − 14.4·27-s + 5·29-s − 3.70·31-s − 0.763·33-s − 3·37-s + 4.00·39-s + 4.76·41-s + 1.76·43-s − 2·47-s + 49-s + 8.00·51-s − 8.47·53-s − 14.4·57-s − 11.7·59-s − 9.70·61-s + 7.47·63-s − 4.23·67-s − 20.1·69-s + ⋯
L(s)  = 1  − 1.86·3-s + 0.377·7-s + 2.49·9-s + 0.0711·11-s − 0.342·13-s − 0.599·17-s + 1.02·19-s − 0.706·21-s + 1.30·23-s − 2.78·27-s + 0.928·29-s − 0.666·31-s − 0.132·33-s − 0.493·37-s + 0.640·39-s + 0.744·41-s + 0.268·43-s − 0.291·47-s + 0.142·49-s + 1.12·51-s − 1.16·53-s − 1.91·57-s − 1.52·59-s − 1.24·61-s + 0.941·63-s − 0.517·67-s − 2.42·69-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9131947241\)
\(L(\frac12)\) \(\approx\) \(0.9131947241\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.23T + 3T^{2} \)
11 \( 1 - 0.236T + 11T^{2} \)
13 \( 1 + 1.23T + 13T^{2} \)
17 \( 1 + 2.47T + 17T^{2} \)
19 \( 1 - 4.47T + 19T^{2} \)
23 \( 1 - 6.23T + 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 3.70T + 31T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 - 4.76T + 41T^{2} \)
43 \( 1 - 1.76T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 + 8.47T + 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 + 9.70T + 61T^{2} \)
67 \( 1 + 4.23T + 67T^{2} \)
71 \( 1 + 8.70T + 71T^{2} \)
73 \( 1 - 8.76T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 + 7.70T + 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 + 5.23T + 97T^{2} \)
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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

−8.997817766239093368584896637960, −7.71636946585161733462585470812, −7.15871599984143801493225201712, −6.40248591351541448346145230388, −5.72578583867714300749631665858, −4.85763268688068697215320293662, −4.57977576161775762894777885259, −3.22105487340315837040061081294, −1.69974817186533828264479277417, −0.67762939405580346289441619077, 0.67762939405580346289441619077, 1.69974817186533828264479277417, 3.22105487340315837040061081294, 4.57977576161775762894777885259, 4.85763268688068697215320293662, 5.72578583867714300749631665858, 6.40248591351541448346145230388, 7.15871599984143801493225201712, 7.71636946585161733462585470812, 8.997817766239093368584896637960

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