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Properties

Label 2-221760-1.1-c1-0-249
Degree $2$
Conductor $221760$
Sign $-1$
Analytic cond. $1770.76$
Root an. cond. $42.0804$
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  − 5-s + 7-s − 11-s + 4·13-s + 4·19-s + 25-s − 6·29-s − 10·31-s − 35-s − 2·37-s + 12·41-s + 4·43-s − 6·47-s + 49-s − 6·53-s + 55-s − 6·59-s + 4·61-s − 4·65-s + 4·67-s − 12·71-s − 4·73-s − 77-s + 8·79-s + 12·83-s − 18·89-s + 4·91-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.301·11-s + 1.10·13-s + 0.917·19-s + 1/5·25-s − 1.11·29-s − 1.79·31-s − 0.169·35-s − 0.328·37-s + 1.87·41-s + 0.609·43-s − 0.875·47-s + 1/7·49-s − 0.824·53-s + 0.134·55-s − 0.781·59-s + 0.512·61-s − 0.496·65-s + 0.488·67-s − 1.42·71-s − 0.468·73-s − 0.113·77-s + 0.900·79-s + 1.31·83-s − 1.90·89-s + 0.419·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221760\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(1770.76\)
Root analytic conductor: \(42.0804\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 221760,\ (\ :1/2),\ -1)\)

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

−13.15050573867477, −12.62005477256112, −12.48278928319188, −11.54563745501218, −11.38419731639499, −10.91907064464937, −10.63149045912937, −9.862588210685182, −9.369068779838807, −8.992415049027091, −8.511524562485516, −7.886324767298142, −7.517017748992711, −7.231011721605541, −6.459970851524146, −5.879913614206708, −5.550231531929266, −4.978348456022852, −4.354968598042529, −3.812560676504704, −3.403517634516216, −2.829725772081891, −2.008342081152069, −1.501071636625442, −0.8149859346699499, 0, 0.8149859346699499, 1.501071636625442, 2.008342081152069, 2.829725772081891, 3.403517634516216, 3.812560676504704, 4.354968598042529, 4.978348456022852, 5.550231531929266, 5.879913614206708, 6.459970851524146, 7.231011721605541, 7.517017748992711, 7.886324767298142, 8.511524562485516, 8.992415049027091, 9.369068779838807, 9.862588210685182, 10.63149045912937, 10.91907064464937, 11.38419731639499, 11.54563745501218, 12.48278928319188, 12.62005477256112, 13.15050573867477

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