Properties

Label 2-7400-1.1-c1-0-122
Degree $2$
Conductor $7400$
Sign $1$
Analytic cond. $59.0892$
Root an. cond. $7.68695$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 5·7-s + 6·9-s − 3·11-s + 4·13-s + 4·17-s − 4·19-s + 15·21-s + 4·23-s + 9·27-s − 8·29-s − 2·31-s − 9·33-s − 37-s + 12·39-s − 5·41-s + 10·43-s − 7·47-s + 18·49-s + 12·51-s − 3·53-s − 12·57-s + 4·59-s + 2·61-s + 30·63-s + 12·67-s + 12·69-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.88·7-s + 2·9-s − 0.904·11-s + 1.10·13-s + 0.970·17-s − 0.917·19-s + 3.27·21-s + 0.834·23-s + 1.73·27-s − 1.48·29-s − 0.359·31-s − 1.56·33-s − 0.164·37-s + 1.92·39-s − 0.780·41-s + 1.52·43-s − 1.02·47-s + 18/7·49-s + 1.68·51-s − 0.412·53-s − 1.58·57-s + 0.520·59-s + 0.256·61-s + 3.77·63-s + 1.46·67-s + 1.44·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7400\)    =    \(2^{3} \cdot 5^{2} \cdot 37\)
Sign: $1$
Analytic conductor: \(59.0892\)
Root analytic conductor: \(7.68695\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.445107284\)
\(L(\frac12)\) \(\approx\) \(5.445107284\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37 \( 1 + T \)
good3 \( 1 - p T + p T^{2} \)
7 \( 1 - 5 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.003007456202266984261202045035, −7.62671177713825196112539495422, −6.81163234190230859454513314465, −5.56507361395207071771924213009, −5.03426634258033597819832419745, −4.08579931981081615421393750565, −3.58343262924882839175263971200, −2.59101230562922757515450419613, −1.90805919805961316113453268552, −1.22084630265444988502766341564, 1.22084630265444988502766341564, 1.90805919805961316113453268552, 2.59101230562922757515450419613, 3.58343262924882839175263971200, 4.08579931981081615421393750565, 5.03426634258033597819832419745, 5.56507361395207071771924213009, 6.81163234190230859454513314465, 7.62671177713825196112539495422, 8.003007456202266984261202045035

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