Properties

Label 2-221760-1.1-c1-0-141
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s − 11-s + 6·13-s − 2·17-s + 6·23-s + 25-s + 6·29-s + 2·31-s − 35-s − 10·37-s + 8·41-s − 8·43-s + 4·47-s + 49-s + 6·53-s + 55-s + 6·59-s + 8·61-s − 6·65-s + 14·67-s − 8·71-s − 2·73-s − 77-s + 16·79-s − 12·83-s + 2·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.301·11-s + 1.66·13-s − 0.485·17-s + 1.25·23-s + 1/5·25-s + 1.11·29-s + 0.359·31-s − 0.169·35-s − 1.64·37-s + 1.24·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.824·53-s + 0.134·55-s + 0.781·59-s + 1.02·61-s − 0.744·65-s + 1.71·67-s − 0.949·71-s − 0.234·73-s − 0.113·77-s + 1.80·79-s − 1.31·83-s + 0.216·85-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: $\chi_{221760} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 221760,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.410677878\)
\(L(\frac12)\) \(\approx\) \(3.410677878\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
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

−13.14745403778527, −12.44567014609653, −11.99358111224930, −11.55554288334278, −10.97386264760782, −10.82291412609893, −10.30214171844431, −9.717705766177276, −9.010897213935255, −8.635334733295752, −8.368574716918791, −7.884734072184706, −7.128467751637603, −6.810108667473933, −6.358016114394092, −5.647782752462836, −5.219167931630219, −4.688807890067347, −4.093033398251657, −3.617418304195966, −3.102024696612711, −2.449621300964400, −1.776338389135085, −1.015555103613020, −0.6143136966475335, 0.6143136966475335, 1.015555103613020, 1.776338389135085, 2.449621300964400, 3.102024696612711, 3.617418304195966, 4.093033398251657, 4.688807890067347, 5.219167931630219, 5.647782752462836, 6.358016114394092, 6.810108667473933, 7.128467751637603, 7.884734072184706, 8.368574716918791, 8.635334733295752, 9.010897213935255, 9.717705766177276, 10.30214171844431, 10.82291412609893, 10.97386264760782, 11.55554288334278, 11.99358111224930, 12.44567014609653, 13.14745403778527

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