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Properties

Label 2-222180-1.1-c1-0-8
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 + 2·11-s + 4·13-s + 15-s − 2·17-s − 2·19-s + 21-s + 25-s − 27-s + 6·29-s − 2·31-s − 2·33-s + 35-s − 10·37-s − 4·39-s − 10·41-s − 12·43-s − 45-s − 8·47-s + 49-s + 2·51-s − 2·55-s + 2·57-s − 8·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 0.258·15-s − 0.485·17-s − 0.458·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.348·33-s + 0.169·35-s − 1.64·37-s − 0.640·39-s − 1.56·41-s − 1.82·43-s − 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.269·55-s + 0.264·57-s − 1.04·59-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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−13.24058081525793, −12.58989196909870, −12.25942165147569, −11.74715759264394, −11.39293183675577, −10.95053228366671, −10.35098162470682, −10.13941627496028, −9.456261582613978, −8.863903862658255, −8.479024311294105, −8.173688877939478, −7.397938966958942, −6.720790230455751, −6.564122915744250, −6.240225565292530, −5.341733439114454, −5.031914777718809, −4.424566967519443, −3.794130284864137, −3.442704446876719, −2.913946624309267, −1.809213557276801, −1.591249957748735, −0.6623149391226936, 0, 0.6623149391226936, 1.591249957748735, 1.809213557276801, 2.913946624309267, 3.442704446876719, 3.794130284864137, 4.424566967519443, 5.031914777718809, 5.341733439114454, 6.240225565292530, 6.564122915744250, 6.720790230455751, 7.397938966958942, 8.173688877939478, 8.479024311294105, 8.863903862658255, 9.456261582613978, 10.13941627496028, 10.35098162470682, 10.95053228366671, 11.39293183675577, 11.74715759264394, 12.25942165147569, 12.58989196909870, 13.24058081525793

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