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Properties

Label 2-177450-1.1-c1-0-148
Degree $2$
Conductor $177450$
Sign $1$
Analytic cond. $1416.94$
Root an. cond. $37.6423$
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  + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s − 11-s + 12-s + 14-s + 16-s + 17-s + 18-s + 19-s + 21-s − 22-s + 6·23-s + 24-s + 27-s + 28-s + 9·29-s + 2·31-s + 32-s − 33-s + 34-s + 36-s + 2·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.229·19-s + 0.218·21-s − 0.213·22-s + 1.25·23-s + 0.204·24-s + 0.192·27-s + 0.188·28-s + 1.67·29-s + 0.359·31-s + 0.176·32-s − 0.174·33-s + 0.171·34-s + 1/6·36-s + 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1416.94\)
Root analytic conductor: \(37.6423\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.084100737\)
\(L(\frac12)\) \(\approx\) \(8.084100737\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
13 \( 1 \)
good11 \( 1 + T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.21069933570678, −12.84740824392080, −12.21731093270007, −11.91926613945210, −11.39499513479396, −10.72964115570001, −10.47825925286135, −9.966825412076666, −9.261316209509432, −8.872315164398326, −8.380905707197957, −7.758403364418896, −7.415974827859895, −6.951220354964471, −6.215162874737410, −5.921945931537939, −5.098237624464919, −4.751019855758026, −4.342180840958599, −3.563888696759501, −3.129863672673226, −2.544950503991200, −2.127992827256403, −1.167433621223349, −0.7735033479693767, 0.7735033479693767, 1.167433621223349, 2.127992827256403, 2.544950503991200, 3.129863672673226, 3.563888696759501, 4.342180840958599, 4.751019855758026, 5.098237624464919, 5.921945931537939, 6.215162874737410, 6.951220354964471, 7.415974827859895, 7.758403364418896, 8.380905707197957, 8.872315164398326, 9.261316209509432, 9.966825412076666, 10.47825925286135, 10.72964115570001, 11.39499513479396, 11.91926613945210, 12.21731093270007, 12.84740824392080, 13.21069933570678

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