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Properties

Label 2-222180-1.1-c1-0-14
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 − 5·13-s + 15-s + 3·17-s + 2·19-s + 21-s + 25-s − 27-s + 29-s + 10·31-s + 35-s + 8·37-s + 5·39-s + 8·41-s + 10·43-s − 45-s − 3·47-s + 49-s − 3·51-s − 2·53-s − 2·57-s − 4·59-s + 6·61-s − 63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.38·13-s + 0.258·15-s + 0.727·17-s + 0.458·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s + 0.185·29-s + 1.79·31-s + 0.169·35-s + 1.31·37-s + 0.800·39-s + 1.24·41-s + 1.52·43-s − 0.149·45-s − 0.437·47-s + 1/7·49-s − 0.420·51-s − 0.274·53-s − 0.264·57-s − 0.520·59-s + 0.768·61-s − 0.125·63-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 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−12.95245837124443, −12.64537091428766, −12.23424996592330, −11.90353324273025, −11.33014627379610, −11.00047247487553, −10.35038956714519, −9.848089835021698, −9.624677401773273, −9.156668245727038, −8.296176577275107, −7.953614192850354, −7.450279155311276, −7.088031176186549, −6.445640572975517, −6.019428159329206, −5.478783339621877, −4.906594351926288, −4.477905230008354, −3.994308755755186, −3.290287529761901, −2.605237838111566, −2.383676936902839, −1.164060670593179, −0.8163592660371966, 0, 0.8163592660371966, 1.164060670593179, 2.383676936902839, 2.605237838111566, 3.290287529761901, 3.994308755755186, 4.477905230008354, 4.906594351926288, 5.478783339621877, 6.019428159329206, 6.445640572975517, 7.088031176186549, 7.450279155311276, 7.953614192850354, 8.296176577275107, 9.156668245727038, 9.624677401773273, 9.848089835021698, 10.35038956714519, 11.00047247487553, 11.33014627379610, 11.90353324273025, 12.23424996592330, 12.64537091428766, 12.95245837124443

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