Properties

Label 2-9200-1.1-c1-0-72
Degree $2$
Conductor $9200$
Sign $1$
Analytic cond. $73.4623$
Root an. cond. $8.57101$
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  + 2·3-s − 7-s + 9-s + 5·11-s + 13-s − 4·17-s − 7·19-s − 2·21-s + 23-s − 4·27-s + 5·29-s − 2·31-s + 10·33-s − 2·37-s + 2·39-s + 11·41-s − 43-s + 8·47-s − 6·49-s − 8·51-s − 14·57-s + 14·59-s + 10·61-s − 63-s + 8·67-s + 2·69-s + 10·71-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s − 0.970·17-s − 1.60·19-s − 0.436·21-s + 0.208·23-s − 0.769·27-s + 0.928·29-s − 0.359·31-s + 1.74·33-s − 0.328·37-s + 0.320·39-s + 1.71·41-s − 0.152·43-s + 1.16·47-s − 6/7·49-s − 1.12·51-s − 1.85·57-s + 1.82·59-s + 1.28·61-s − 0.125·63-s + 0.977·67-s + 0.240·69-s + 1.18·71-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: yes
Arithmetic: yes
Character: $\chi_{9200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.116023106\)
\(L(\frac12)\) \(\approx\) \(3.116023106\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−7.891692787276342096834772565072, −6.89683672326933800534947165278, −6.56191559535968493178322265131, −5.85320262180626887736682335254, −4.70877365547697582315761730393, −3.92338595003306067616288650284, −3.62349190950048953286603490312, −2.48078906059132474481242328022, −2.05191779631866543865602232431, −0.793767924709225545348060160981, 0.793767924709225545348060160981, 2.05191779631866543865602232431, 2.48078906059132474481242328022, 3.62349190950048953286603490312, 3.92338595003306067616288650284, 4.70877365547697582315761730393, 5.85320262180626887736682335254, 6.56191559535968493178322265131, 6.89683672326933800534947165278, 7.891692787276342096834772565072

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