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Properties

Label 2-222180-1.1-c1-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 7-s + 9-s − 2·11-s − 2·13-s + 15-s − 6·17-s − 4·19-s + 21-s + 25-s + 27-s − 6·29-s − 2·31-s − 2·33-s + 35-s − 6·37-s − 2·39-s + 12·41-s − 4·43-s + 45-s + 49-s − 6·51-s − 6·53-s − 2·55-s − 4·57-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 − 0.554·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-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 − 0.986·37-s − 0.320·39-s + 1.87·41-s − 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 0.269·55-s − 0.529·57-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: \(0\)
Selberg data: \((2,\ 222180,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.209102254\)
\(L(\frac12)\) \(\approx\) \(1.209102254\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−13.09847226516315, −12.54363509167568, −12.23640829139272, −11.42781603642325, −10.97483254083661, −10.72597105047047, −10.16944381949333, −9.601622205913069, −9.073221952901296, −8.920526517460527, −8.173492053874277, −7.866389017487434, −7.246602611076523, −6.849265209097456, −6.275879869612026, −5.716253646229109, −5.176849158701245, −4.583838815570715, −4.260218393793564, −3.570132827478020, −2.898739342575902, −2.272432582175354, −2.036185002565656, −1.361217322783023, −0.2715242284922175, 0.2715242284922175, 1.361217322783023, 2.036185002565656, 2.272432582175354, 2.898739342575902, 3.570132827478020, 4.260218393793564, 4.583838815570715, 5.176849158701245, 5.716253646229109, 6.275879869612026, 6.849265209097456, 7.246602611076523, 7.866389017487434, 8.173492053874277, 8.920526517460527, 9.073221952901296, 9.601622205913069, 10.16944381949333, 10.72597105047047, 10.97483254083661, 11.42781603642325, 12.23640829139272, 12.54363509167568, 13.09847226516315

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