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Properties

Label 2-59850-1.1-c1-0-101
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 7-s − 8-s + 2·11-s − 5·13-s − 14-s + 16-s − 2·17-s + 19-s − 2·22-s − 4·23-s + 5·26-s + 28-s + 4·31-s − 32-s + 2·34-s − 2·37-s − 38-s + 7·41-s − 12·43-s + 2·44-s + 4·46-s − 13·47-s + 49-s − 5·52-s − 56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.603·11-s − 1.38·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.229·19-s − 0.426·22-s − 0.834·23-s + 0.980·26-s + 0.188·28-s + 0.718·31-s − 0.176·32-s + 0.342·34-s − 0.328·37-s − 0.162·38-s + 1.09·41-s − 1.82·43-s + 0.301·44-s + 0.589·46-s − 1.89·47-s + 1/7·49-s − 0.693·52-s − 0.133·56-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: \(1\)
Selberg data: \((2,\ 59850,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
19 \( 1 - T \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.69099339744091, −14.19480896754279, −13.58518475788185, −13.03329188366098, −12.34952693142361, −11.87790943506775, −11.63679146627489, −10.96862824691737, −10.41076592519076, −9.736340275196454, −9.664344944147632, −8.886850411348589, −8.361017216123621, −7.857088623264217, −7.388298540054859, −6.705892577503922, −6.377138969153722, −5.609948391965677, −4.849006443832038, −4.553739355143356, −3.625952201447980, −3.033082117026574, −2.154350979524372, −1.823769136716791, −0.8391925717661824, 0, 0.8391925717661824, 1.823769136716791, 2.154350979524372, 3.033082117026574, 3.625952201447980, 4.553739355143356, 4.849006443832038, 5.609948391965677, 6.377138969153722, 6.705892577503922, 7.388298540054859, 7.857088623264217, 8.361017216123621, 8.886850411348589, 9.664344944147632, 9.736340275196454, 10.41076592519076, 10.96862824691737, 11.63679146627489, 11.87790943506775, 12.34952693142361, 13.03329188366098, 13.58518475788185, 14.19480896754279, 14.69099339744091

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