Properties

Label 2-9200-1.1-c1-0-168
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  − 3.21·3-s + 2.43·7-s + 7.36·9-s + 0.884·11-s + 5.10·13-s + 0.366·17-s + 2.79·19-s − 7.82·21-s − 23-s − 14.0·27-s + 8.02·29-s − 7.24·31-s − 2.84·33-s − 3.10·37-s − 16.4·39-s − 3.47·41-s − 8.56·43-s − 5.25·47-s − 1.08·49-s − 1.18·51-s − 11.6·53-s − 9.00·57-s + 9.33·59-s + 5.46·61-s + 17.9·63-s − 1.49·67-s + 3.21·69-s + ⋯
L(s)  = 1  − 1.85·3-s + 0.919·7-s + 2.45·9-s + 0.266·11-s + 1.41·13-s + 0.0889·17-s + 0.642·19-s − 1.70·21-s − 0.208·23-s − 2.70·27-s + 1.49·29-s − 1.30·31-s − 0.495·33-s − 0.509·37-s − 2.63·39-s − 0.542·41-s − 1.30·43-s − 0.766·47-s − 0.155·49-s − 0.165·51-s − 1.59·53-s − 1.19·57-s + 1.21·59-s + 0.699·61-s + 2.25·63-s − 0.182·67-s + 0.387·69-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 \( 1 + 3.21T + 3T^{2} \)
7 \( 1 - 2.43T + 7T^{2} \)
11 \( 1 - 0.884T + 11T^{2} \)
13 \( 1 - 5.10T + 13T^{2} \)
17 \( 1 - 0.366T + 17T^{2} \)
19 \( 1 - 2.79T + 19T^{2} \)
29 \( 1 - 8.02T + 29T^{2} \)
31 \( 1 + 7.24T + 31T^{2} \)
37 \( 1 + 3.10T + 37T^{2} \)
41 \( 1 + 3.47T + 41T^{2} \)
43 \( 1 + 8.56T + 43T^{2} \)
47 \( 1 + 5.25T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 - 9.33T + 59T^{2} \)
61 \( 1 - 5.46T + 61T^{2} \)
67 \( 1 + 1.49T + 67T^{2} \)
71 \( 1 + 8.29T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + 6.06T + 79T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 + 17.6T + 89T^{2} \)
97 \( 1 + 6.55T + 97T^{2} \)
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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

−7.01035879411378867504852568997, −6.71368957805904267581834540510, −5.84511327947992045627301646948, −5.44342171730170351835234321716, −4.74037932256767211890878986408, −4.13909410328967590114284806697, −3.23629049982042394139127940172, −1.56246571210034551652050769703, −1.30294505355202260070691875945, 0, 1.30294505355202260070691875945, 1.56246571210034551652050769703, 3.23629049982042394139127940172, 4.13909410328967590114284806697, 4.74037932256767211890878986408, 5.44342171730170351835234321716, 5.84511327947992045627301646948, 6.71368957805904267581834540510, 7.01035879411378867504852568997

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