Properties

Label 2-369600-1.1-c1-0-118
Degree $2$
Conductor $369600$
Sign $1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
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-s − 7-s + 9-s + 11-s − 6·13-s − 2·17-s − 21-s + 6·23-s + 27-s − 6·29-s + 2·31-s + 33-s + 10·37-s − 6·39-s − 8·41-s + 8·43-s + 4·47-s + 49-s − 2·51-s + 6·53-s − 6·59-s + 8·61-s − 63-s − 14·67-s + 6·69-s + 8·71-s + 2·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s − 0.485·17-s − 0.218·21-s + 1.25·23-s + 0.192·27-s − 1.11·29-s + 0.359·31-s + 0.174·33-s + 1.64·37-s − 0.960·39-s − 1.24·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.781·59-s + 1.02·61-s − 0.125·63-s − 1.71·67-s + 0.722·69-s + 0.949·71-s + 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{369600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.534021007\)
\(L(\frac12)\) \(\approx\) \(2.534021007\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
good13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−12.66080220818633, −12.13477988140687, −11.56840158635998, −11.21609832030569, −10.63511509491298, −10.16059168553373, −9.629314199938369, −9.406553096104161, −8.930448772615488, −8.497794638577551, −7.828731833195721, −7.388092803489339, −7.123257610805798, −6.613062343633966, −6.016017384718331, −5.471114747856456, −4.913618103561983, −4.456616413000698, −4.005522932169713, −3.338977250712673, −2.816179747270778, −2.395719932302406, −1.900928832718796, −1.051958538883940, −0.4228667038789291, 0.4228667038789291, 1.051958538883940, 1.900928832718796, 2.395719932302406, 2.816179747270778, 3.338977250712673, 4.005522932169713, 4.456616413000698, 4.913618103561983, 5.471114747856456, 6.016017384718331, 6.613062343633966, 7.123257610805798, 7.388092803489339, 7.828731833195721, 8.497794638577551, 8.930448772615488, 9.406553096104161, 9.629314199938369, 10.16059168553373, 10.63511509491298, 11.21609832030569, 11.56840158635998, 12.13477988140687, 12.66080220818633

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