Properties

Label 2-9200-1.1-c1-0-195
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  + 1.79·3-s + 2.79·7-s + 0.208·9-s − 3.79·11-s − 1.20·13-s + 3.79·17-s − 1.20·19-s + 5·21-s + 23-s − 5.00·27-s − 1.58·29-s − 10.3·31-s − 6.79·33-s + 4·37-s − 2.16·39-s − 2.20·41-s − 7.16·43-s − 13.5·47-s + 0.791·49-s + 6.79·51-s − 6·53-s − 2.16·57-s + 4.41·59-s − 3.37·61-s + 0.582·63-s − 7.16·67-s + 1.79·69-s + ⋯
L(s)  = 1  + 1.03·3-s + 1.05·7-s + 0.0695·9-s − 1.14·11-s − 0.335·13-s + 0.919·17-s − 0.277·19-s + 1.09·21-s + 0.208·23-s − 0.962·27-s − 0.293·29-s − 1.86·31-s − 1.18·33-s + 0.657·37-s − 0.346·39-s − 0.344·41-s − 1.09·43-s − 1.98·47-s + 0.113·49-s + 0.950·51-s − 0.824·53-s − 0.286·57-s + 0.575·59-s − 0.431·61-s + 0.0733·63-s − 0.875·67-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 \( 1 - 1.79T + 3T^{2} \)
7 \( 1 - 2.79T + 7T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
13 \( 1 + 1.20T + 13T^{2} \)
17 \( 1 - 3.79T + 17T^{2} \)
19 \( 1 + 1.20T + 19T^{2} \)
29 \( 1 + 1.58T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 2.20T + 41T^{2} \)
43 \( 1 + 7.16T + 43T^{2} \)
47 \( 1 + 13.5T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 4.41T + 59T^{2} \)
61 \( 1 + 3.37T + 61T^{2} \)
67 \( 1 + 7.16T + 67T^{2} \)
71 \( 1 - 5.37T + 71T^{2} \)
73 \( 1 - 14.7T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 3.16T + 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−7.69071184685469354061294145647, −6.95170271766643310468278822068, −5.84898395945617048176919807806, −5.21375180861537758944244636791, −4.68841296474468421771286576864, −3.60251565526635849271839419260, −3.07164535779759277953204527935, −2.17650259031641557046713767375, −1.57142420745204458670775614313, 0, 1.57142420745204458670775614313, 2.17650259031641557046713767375, 3.07164535779759277953204527935, 3.60251565526635849271839419260, 4.68841296474468421771286576864, 5.21375180861537758944244636791, 5.84898395945617048176919807806, 6.95170271766643310468278822068, 7.69071184685469354061294145647

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