Properties

Label 2-9200-1.1-c1-0-116
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  + 3.30·3-s + 0.852·7-s + 7.94·9-s − 1.80·11-s + 2.70·13-s + 4.05·17-s − 7.19·19-s + 2.82·21-s − 23-s + 16.3·27-s − 3.76·29-s − 4.00·31-s − 5.98·33-s + 11.4·37-s + 8.95·39-s + 6.09·41-s + 11.4·43-s − 7.31·47-s − 6.27·49-s + 13.4·51-s + 0.406·53-s − 23.8·57-s + 4.83·59-s + 9.22·61-s + 6.77·63-s + 9.64·67-s − 3.30·69-s + ⋯
L(s)  = 1  + 1.91·3-s + 0.322·7-s + 2.64·9-s − 0.545·11-s + 0.750·13-s + 0.982·17-s − 1.65·19-s + 0.615·21-s − 0.208·23-s + 3.15·27-s − 0.698·29-s − 0.719·31-s − 1.04·33-s + 1.88·37-s + 1.43·39-s + 0.951·41-s + 1.73·43-s − 1.06·47-s − 0.896·49-s + 1.87·51-s + 0.0558·53-s − 3.15·57-s + 0.629·59-s + 1.18·61-s + 0.854·63-s + 1.17·67-s − 0.398·69-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

\(L(1)\) \(\approx\) \(5.053008124\)
\(L(\frac12)\) \(\approx\) \(5.053008124\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.30T + 3T^{2} \)
7 \( 1 - 0.852T + 7T^{2} \)
11 \( 1 + 1.80T + 11T^{2} \)
13 \( 1 - 2.70T + 13T^{2} \)
17 \( 1 - 4.05T + 17T^{2} \)
19 \( 1 + 7.19T + 19T^{2} \)
29 \( 1 + 3.76T + 29T^{2} \)
31 \( 1 + 4.00T + 31T^{2} \)
37 \( 1 - 11.4T + 37T^{2} \)
41 \( 1 - 6.09T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + 7.31T + 47T^{2} \)
53 \( 1 - 0.406T + 53T^{2} \)
59 \( 1 - 4.83T + 59T^{2} \)
61 \( 1 - 9.22T + 61T^{2} \)
67 \( 1 - 9.64T + 67T^{2} \)
71 \( 1 - 7.31T + 71T^{2} \)
73 \( 1 - 5.46T + 73T^{2} \)
79 \( 1 + 8.60T + 79T^{2} \)
83 \( 1 + 2.49T + 83T^{2} \)
89 \( 1 + 5.41T + 89T^{2} \)
97 \( 1 - 8.41T + 97T^{2} \)
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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

−7.84060270990353733567599023254, −7.41055169736788669517361242120, −6.48630132191834754099298491317, −5.71269643932048171006779750547, −4.64740671365960861588035796992, −3.98555254112846557428743533293, −3.47184931286659917478078357141, −2.51404333160107399457836256213, −2.06053923406690753615954592791, −1.02982472899499616606146622914, 1.02982472899499616606146622914, 2.06053923406690753615954592791, 2.51404333160107399457836256213, 3.47184931286659917478078357141, 3.98555254112846557428743533293, 4.64740671365960861588035796992, 5.71269643932048171006779750547, 6.48630132191834754099298491317, 7.41055169736788669517361242120, 7.84060270990353733567599023254

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