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Properties

Label 2-222180-1.1-c1-0-20
Degree $2$
Conductor $222180$
Sign $-1$
Analytic cond. $1774.11$
Root an. cond. $42.1202$
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  + 3-s − 5-s + 7-s + 9-s + 3·11-s + 5·13-s − 15-s − 6·17-s − 19-s + 21-s + 25-s + 27-s + 5·31-s + 3·33-s − 35-s + 2·37-s + 5·39-s − 6·41-s − 43-s − 45-s − 12·47-s + 49-s − 6·51-s + 6·53-s − 3·55-s − 57-s + 12·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.38·13-s − 0.258·15-s − 1.45·17-s − 0.229·19-s + 0.218·21-s + 1/5·25-s + 0.192·27-s + 0.898·31-s + 0.522·33-s − 0.169·35-s + 0.328·37-s + 0.800·39-s − 0.937·41-s − 0.152·43-s − 0.149·45-s − 1.75·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s − 0.404·55-s − 0.132·57-s + 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 222180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(222180\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(1774.11\)
Root analytic conductor: \(42.1202\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 222180,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 \)
good11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
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

−13.29341187544018, −12.91757778793569, −12.14878913841476, −11.77039900443378, −11.30520148809754, −10.98674516761409, −10.48279942927032, −9.810699311622477, −9.417307551935215, −8.777825269032258, −8.448542353421437, −8.256608774040746, −7.605241591266897, −6.817635180281425, −6.600498303428310, −6.258432271092729, −5.351804189403813, −4.878154418081543, −4.239716467452784, −3.819363961354918, −3.529806250096666, −2.655498323167215, −2.173466569694214, −1.432513043436949, −0.9873229092256220, 0, 0.9873229092256220, 1.432513043436949, 2.173466569694214, 2.655498323167215, 3.529806250096666, 3.819363961354918, 4.239716467452784, 4.878154418081543, 5.351804189403813, 6.258432271092729, 6.600498303428310, 6.817635180281425, 7.605241591266897, 8.256608774040746, 8.448542353421437, 8.777825269032258, 9.417307551935215, 9.810699311622477, 10.48279942927032, 10.98674516761409, 11.30520148809754, 11.77039900443378, 12.14878913841476, 12.91757778793569, 13.29341187544018

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