Properties

Label 2-97020-1.1-c1-0-50
Degree $2$
Conductor $97020$
Sign $-1$
Analytic cond. $774.708$
Root an. cond. $27.8335$
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  + 5-s − 11-s − 13-s − 6·17-s − 2·19-s + 7·23-s + 25-s − 5·29-s − 4·31-s − 10·37-s − 9·41-s + 3·43-s + 9·47-s + 9·53-s − 55-s + 12·59-s − 65-s + 8·67-s − 2·71-s + 2·73-s + 6·79-s + 4·83-s − 6·85-s − 6·89-s − 2·95-s + 16·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.301·11-s − 0.277·13-s − 1.45·17-s − 0.458·19-s + 1.45·23-s + 1/5·25-s − 0.928·29-s − 0.718·31-s − 1.64·37-s − 1.40·41-s + 0.457·43-s + 1.31·47-s + 1.23·53-s − 0.134·55-s + 1.56·59-s − 0.124·65-s + 0.977·67-s − 0.237·71-s + 0.234·73-s + 0.675·79-s + 0.439·83-s − 0.650·85-s − 0.635·89-s − 0.205·95-s + 1.62·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97020\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(774.708\)
Root analytic conductor: \(27.8335\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 97020,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 + T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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

−13.95822057860431, −13.36521024273742, −13.20680751538865, −12.67275495527237, −12.09881239857043, −11.55876112218026, −10.95040275760582, −10.64145351911385, −10.21611610707490, −9.433291435697402, −9.067183851570042, −8.664840755283434, −8.148099362713731, −7.294427840212711, −6.899117999027269, −6.662540340707486, −5.714561096645296, −5.331221386111146, −4.891692092815169, −4.120668278794605, −3.638884742645462, −2.863032095889707, −2.196894497562980, −1.844094936398755, −0.8249149584450379, 0, 0.8249149584450379, 1.844094936398755, 2.196894497562980, 2.863032095889707, 3.638884742645462, 4.120668278794605, 4.891692092815169, 5.331221386111146, 5.714561096645296, 6.662540340707486, 6.899117999027269, 7.294427840212711, 8.148099362713731, 8.664840755283434, 9.067183851570042, 9.433291435697402, 10.21611610707490, 10.64145351911385, 10.95040275760582, 11.55876112218026, 12.09881239857043, 12.67275495527237, 13.20680751538865, 13.36521024273742, 13.95822057860431

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