Properties

Label 2-9200-1.1-c1-0-117
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.78·3-s − 1.75·7-s + 0.192·9-s + 4.77·11-s + 1.72·13-s − 7.81·17-s + 2.43·19-s + 3.13·21-s + 23-s + 5.01·27-s + 7.86·29-s − 6.14·31-s − 8.53·33-s − 6.83·37-s − 3.09·39-s − 2.50·41-s + 3.26·43-s − 8.46·47-s − 3.91·49-s + 13.9·51-s − 2.76·53-s − 4.35·57-s − 1.91·59-s − 3.50·61-s − 0.337·63-s + 12.7·67-s − 1.78·69-s + ⋯
L(s)  = 1  − 1.03·3-s − 0.663·7-s + 0.0640·9-s + 1.43·11-s + 0.479·13-s − 1.89·17-s + 0.559·19-s + 0.684·21-s + 0.208·23-s + 0.965·27-s + 1.46·29-s − 1.10·31-s − 1.48·33-s − 1.12·37-s − 0.494·39-s − 0.391·41-s + 0.498·43-s − 1.23·47-s − 0.559·49-s + 1.95·51-s − 0.379·53-s − 0.577·57-s − 0.249·59-s − 0.448·61-s − 0.0424·63-s + 1.55·67-s − 0.215·69-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: $\chi_{9200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.78T + 3T^{2} \)
7 \( 1 + 1.75T + 7T^{2} \)
11 \( 1 - 4.77T + 11T^{2} \)
13 \( 1 - 1.72T + 13T^{2} \)
17 \( 1 + 7.81T + 17T^{2} \)
19 \( 1 - 2.43T + 19T^{2} \)
29 \( 1 - 7.86T + 29T^{2} \)
31 \( 1 + 6.14T + 31T^{2} \)
37 \( 1 + 6.83T + 37T^{2} \)
41 \( 1 + 2.50T + 41T^{2} \)
43 \( 1 - 3.26T + 43T^{2} \)
47 \( 1 + 8.46T + 47T^{2} \)
53 \( 1 + 2.76T + 53T^{2} \)
59 \( 1 + 1.91T + 59T^{2} \)
61 \( 1 + 3.50T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - 0.0111T + 73T^{2} \)
79 \( 1 - 16.6T + 79T^{2} \)
83 \( 1 - 2.64T + 83T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 - 15.8T + 97T^{2} \)
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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

−6.96045146308441326760506769532, −6.45256588683120639054739118946, −6.38430310791472672033273671061, −5.28815678327407983624433286901, −4.73209051014698711712758193809, −3.85938064255510724539618704741, −3.20609155288029770278052061937, −2.06027542982969031802288120685, −1.03107224770745423223291503661, 0, 1.03107224770745423223291503661, 2.06027542982969031802288120685, 3.20609155288029770278052061937, 3.85938064255510724539618704741, 4.73209051014698711712758193809, 5.28815678327407983624433286901, 6.38430310791472672033273671061, 6.45256588683120639054739118946, 6.96045146308441326760506769532

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