↑

Properties

Label 2-222180-1.1-c1-0-23
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 7-s + 9-s − 11-s + 13-s + 15-s + 2·17-s + 5·19-s − 21-s + 25-s + 27-s + 8·29-s − 31-s − 33-s − 35-s + 2·37-s + 39-s − 2·41-s + 43-s + 45-s − 12·47-s + 49-s + 2·51-s + 2·53-s − 55-s + 5·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.258·15-s + 0.485·17-s + 1.14·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s + 1.48·29-s − 0.179·31-s − 0.174·33-s − 0.169·35-s + 0.328·37-s + 0.160·39-s − 0.312·41-s + 0.152·43-s + 0.149·45-s − 1.75·47-s + 1/7·49-s + 0.280·51-s + 0.274·53-s − 0.134·55-s + 0.662·57-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 \( 1 + T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 17 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.39411638462670, −12.71004549243074, −12.30363598274218, −11.99661308483996, −11.12657735267701, −11.00750673965106, −10.10198707367577, −9.955937833077158, −9.560176066586593, −8.952938164416118, −8.542760299552632, −7.916649047924810, −7.684167333700568, −6.953027196974763, −6.536544479347323, −6.054530613264474, −5.443366825264651, −4.931079945061871, −4.506821068441873, −3.582901895849652, −3.368120612337405, −2.721530721721068, −2.271508942480565, −1.382776821374400, −1.029666119704925, 0, 1.029666119704925, 1.382776821374400, 2.271508942480565, 2.721530721721068, 3.368120612337405, 3.582901895849652, 4.506821068441873, 4.931079945061871, 5.443366825264651, 6.054530613264474, 6.536544479347323, 6.953027196974763, 7.684167333700568, 7.916649047924810, 8.542760299552632, 8.952938164416118, 9.560176066586593, 9.955937833077158, 10.10198707367577, 11.00750673965106, 11.12657735267701, 11.99661308483996, 12.30363598274218, 12.71004549243074, 13.39411638462670

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