Properties

Label 2-9200-1.1-c1-0-114
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  + 2.69·3-s + 2.81·7-s + 4.26·9-s + 1.84·11-s − 6.49·13-s + 7.06·17-s + 0.252·19-s + 7.59·21-s + 23-s + 3.42·27-s + 4.12·29-s − 3.54·31-s + 4.98·33-s − 7.91·37-s − 17.5·39-s + 6.53·41-s − 4.93·43-s + 0.851·47-s + 0.935·49-s + 19.0·51-s + 12.5·53-s + 0.680·57-s + 10.3·59-s + 1.01·61-s + 12.0·63-s + 3.37·67-s + 2.69·69-s + ⋯
L(s)  = 1  + 1.55·3-s + 1.06·7-s + 1.42·9-s + 0.557·11-s − 1.80·13-s + 1.71·17-s + 0.0578·19-s + 1.65·21-s + 0.208·23-s + 0.658·27-s + 0.765·29-s − 0.637·31-s + 0.867·33-s − 1.30·37-s − 2.80·39-s + 1.02·41-s − 0.753·43-s + 0.124·47-s + 0.133·49-s + 2.66·51-s + 1.72·53-s + 0.0901·57-s + 1.34·59-s + 0.129·61-s + 1.51·63-s + 0.412·67-s + 0.324·69-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

\(L(1)\) \(\approx\) \(4.662529950\)
\(L(\frac12)\) \(\approx\) \(4.662529950\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.69T + 3T^{2} \)
7 \( 1 - 2.81T + 7T^{2} \)
11 \( 1 - 1.84T + 11T^{2} \)
13 \( 1 + 6.49T + 13T^{2} \)
17 \( 1 - 7.06T + 17T^{2} \)
19 \( 1 - 0.252T + 19T^{2} \)
29 \( 1 - 4.12T + 29T^{2} \)
31 \( 1 + 3.54T + 31T^{2} \)
37 \( 1 + 7.91T + 37T^{2} \)
41 \( 1 - 6.53T + 41T^{2} \)
43 \( 1 + 4.93T + 43T^{2} \)
47 \( 1 - 0.851T + 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 1.01T + 61T^{2} \)
67 \( 1 - 3.37T + 67T^{2} \)
71 \( 1 - 0.851T + 71T^{2} \)
73 \( 1 - 9.75T + 73T^{2} \)
79 \( 1 - 16.5T + 79T^{2} \)
83 \( 1 - 0.696T + 83T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 - 5.48T + 97T^{2} \)
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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

−7.65521560336174936375256079098, −7.45086275933962655790180056487, −6.63024338726410942326475894240, −5.31690805439237261910918843550, −5.02349614764919735297816244341, −3.99939278361304378790067540245, −3.43069882606569333689221329595, −2.53276216421660279954819919468, −1.98280332384781333034854822044, −1.00879918945531840730546873230, 1.00879918945531840730546873230, 1.98280332384781333034854822044, 2.53276216421660279954819919468, 3.43069882606569333689221329595, 3.99939278361304378790067540245, 5.02349614764919735297816244341, 5.31690805439237261910918843550, 6.63024338726410942326475894240, 7.45086275933962655790180056487, 7.65521560336174936375256079098

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