Properties

Label 2-9200-1.1-c1-0-70
Degree $2$
Conductor $9200$
Sign $1$
Analytic cond. $73.4623$
Root an. cond. $8.57101$
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  − 2.95·3-s + 3.95·7-s + 5.74·9-s + 0.957·11-s + 2.74·13-s − 5.74·17-s + 6.74·19-s − 11.7·21-s + 23-s − 8.12·27-s + 5.21·29-s + 5.95·31-s − 2.83·33-s − 9.12·37-s − 8.12·39-s + 0.252·41-s + 8·43-s + 5.49·47-s + 8.66·49-s + 16.9·51-s + 7.12·53-s − 19.9·57-s + 4.78·59-s + 12.4·61-s + 22.7·63-s + 9.12·67-s − 2.95·69-s + ⋯
L(s)  = 1  − 1.70·3-s + 1.49·7-s + 1.91·9-s + 0.288·11-s + 0.761·13-s − 1.39·17-s + 1.54·19-s − 2.55·21-s + 0.208·23-s − 1.56·27-s + 0.967·29-s + 1.07·31-s − 0.493·33-s − 1.50·37-s − 1.30·39-s + 0.0394·41-s + 1.21·43-s + 0.801·47-s + 1.23·49-s + 2.38·51-s + 0.978·53-s − 2.64·57-s + 0.623·59-s + 1.59·61-s + 2.86·63-s + 1.11·67-s − 0.356·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9200\)    =    \(2^{4} \cdot 5^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(73.4623\)
Root analytic conductor: \(8.57101\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.651440883\)
\(L(\frac12)\) \(\approx\) \(1.651440883\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( 1 - T \)
good3 \( 1 + 2.95T + 3T^{2} \)
7 \( 1 - 3.95T + 7T^{2} \)
11 \( 1 - 0.957T + 11T^{2} \)
13 \( 1 - 2.74T + 13T^{2} \)
17 \( 1 + 5.74T + 17T^{2} \)
19 \( 1 - 6.74T + 19T^{2} \)
29 \( 1 - 5.21T + 29T^{2} \)
31 \( 1 - 5.95T + 31T^{2} \)
37 \( 1 + 9.12T + 37T^{2} \)
41 \( 1 - 0.252T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 5.49T + 47T^{2} \)
53 \( 1 - 7.12T + 53T^{2} \)
59 \( 1 - 4.78T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 - 9.12T + 67T^{2} \)
71 \( 1 + 1.66T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 - 0.704T + 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−7.39626459211989044649129353885, −7.04461633411761207641521195176, −6.21744034304551447475109766317, −5.63154062862686269885341364263, −4.97117943518970933271121890751, −4.54438445145146960783571510516, −3.77571440525034708741688765385, −2.39726588774013309314386967935, −1.33831345209893637612313925070, −0.791786886986594595732976553433, 0.791786886986594595732976553433, 1.33831345209893637612313925070, 2.39726588774013309314386967935, 3.77571440525034708741688765385, 4.54438445145146960783571510516, 4.97117943518970933271121890751, 5.63154062862686269885341364263, 6.21744034304551447475109766317, 7.04461633411761207641521195176, 7.39626459211989044649129353885

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