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Properties

Label 2-59850-1.1-c1-0-17
Degree $2$
Conductor $59850$
Sign $1$
Analytic cond. $477.904$
Root an. cond. $21.8610$
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 + 4-s − 7-s − 8-s − 2·11-s + 3·13-s + 14-s + 16-s − 3·17-s − 19-s + 2·22-s + 23-s − 3·26-s − 28-s − 3·29-s + 31-s − 32-s + 3·34-s − 8·37-s + 38-s + 11·41-s + 11·43-s − 2·44-s − 46-s − 6·47-s + 49-s + 3·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.603·11-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.229·19-s + 0.426·22-s + 0.208·23-s − 0.588·26-s − 0.188·28-s − 0.557·29-s + 0.179·31-s − 0.176·32-s + 0.514·34-s − 1.31·37-s + 0.162·38-s + 1.71·41-s + 1.67·43-s − 0.301·44-s − 0.147·46-s − 0.875·47-s + 1/7·49-s + 0.416·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(59850\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(477.904\)
Root analytic conductor: \(21.8610\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 59850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.216184378\)
\(L(\frac12)\) \(\approx\) \(1.216184378\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
19 \( 1 + T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−14.34995096375576, −13.73754946750086, −13.29851474095692, −12.66542679570786, −12.46468952159007, −11.53868864563425, −11.19914791465884, −10.69994717840679, −10.27955915805765, −9.672475084396328, −9.043864507195593, −8.766000835652800, −8.184194369084590, −7.464141854097587, −7.212009542989539, −6.301439693377802, −6.126174411106517, −5.350139773264321, −4.715101862429444, −3.883537754809768, −3.451069413092793, −2.527471532074180, −2.165547327001336, −1.192631839249640, −0.4544111024925486, 0.4544111024925486, 1.192631839249640, 2.165547327001336, 2.527471532074180, 3.451069413092793, 3.883537754809768, 4.715101862429444, 5.350139773264321, 6.126174411106517, 6.301439693377802, 7.212009542989539, 7.464141854097587, 8.184194369084590, 8.766000835652800, 9.043864507195593, 9.672475084396328, 10.27955915805765, 10.69994717840679, 11.19914791465884, 11.53868864563425, 12.46468952159007, 12.66542679570786, 13.29851474095692, 13.73754946750086, 14.34995096375576

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