Properties

Label 2-2800-1.1-c1-0-3
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  − 2.44·3-s + 7-s + 2.99·9-s − 4.89·11-s − 0.449·13-s + 2·17-s − 6.44·19-s − 2.44·21-s + 6.89·23-s − 2.89·29-s + 0.898·31-s + 11.9·33-s + 2·37-s + 1.10·39-s − 10.8·41-s − 8.89·43-s + 0.898·47-s + 49-s − 4.89·51-s − 1.10·53-s + 15.7·57-s + 6.44·59-s + 8.44·61-s + 2.99·63-s + 8·67-s − 16.8·69-s + 10.8·71-s + ⋯
L(s)  = 1  − 1.41·3-s + 0.377·7-s + 0.999·9-s − 1.47·11-s − 0.124·13-s + 0.485·17-s − 1.47·19-s − 0.534·21-s + 1.43·23-s − 0.538·29-s + 0.161·31-s + 2.08·33-s + 0.328·37-s + 0.176·39-s − 1.70·41-s − 1.35·43-s + 0.131·47-s + 0.142·49-s − 0.685·51-s − 0.151·53-s + 2.09·57-s + 0.839·59-s + 1.08·61-s + 0.377·63-s + 0.977·67-s − 2.03·69-s + 1.29·71-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7153896710\)
\(L(\frac12)\) \(\approx\) \(0.7153896710\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.44T + 3T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 + 0.449T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 6.44T + 19T^{2} \)
23 \( 1 - 6.89T + 23T^{2} \)
29 \( 1 + 2.89T + 29T^{2} \)
31 \( 1 - 0.898T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 8.89T + 43T^{2} \)
47 \( 1 - 0.898T + 47T^{2} \)
53 \( 1 + 1.10T + 53T^{2} \)
59 \( 1 - 6.44T + 59T^{2} \)
61 \( 1 - 8.44T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + 6.89T + 73T^{2} \)
79 \( 1 - 2.89T + 79T^{2} \)
83 \( 1 + 2.44T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 3.79T + 97T^{2} \)
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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

−8.615721155881233092428881639572, −8.081080887741216134287357995905, −7.05448023130479055113302022373, −6.53612297344938926749316659366, −5.42813493035793523883578306300, −5.22082619980247530919039799153, −4.37192271673031442501886828903, −3.09959504965922605075712127398, −1.94698194085831653681081679505, −0.54798195217476330134891053710, 0.54798195217476330134891053710, 1.94698194085831653681081679505, 3.09959504965922605075712127398, 4.37192271673031442501886828903, 5.22082619980247530919039799153, 5.42813493035793523883578306300, 6.53612297344938926749316659366, 7.05448023130479055113302022373, 8.081080887741216134287357995905, 8.615721155881233092428881639572

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