Properties

Label 2-97020-1.1-c1-0-39
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 11-s − 2·13-s − 7·17-s − 3·19-s − 3·23-s + 25-s + 5·29-s + 2·37-s + 43-s + 8·47-s + 9·53-s − 55-s − 9·59-s − 5·61-s + 2·65-s − 2·67-s + 12·71-s − 14·79-s + 9·83-s + 7·85-s + 5·89-s + 3·95-s − 7·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 0.554·13-s − 1.69·17-s − 0.688·19-s − 0.625·23-s + 1/5·25-s + 0.928·29-s + 0.328·37-s + 0.152·43-s + 1.16·47-s + 1.23·53-s − 0.134·55-s − 1.17·59-s − 0.640·61-s + 0.248·65-s − 0.244·67-s + 1.42·71-s − 1.57·79-s + 0.987·83-s + 0.759·85-s + 0.529·89-s + 0.307·95-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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: $\chi_{97020} (1, \cdot )$
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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
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

−13.92209181688953, −13.63756382656552, −13.03044951056217, −12.46538042041295, −12.14187041615984, −11.59112170058487, −11.08683967984753, −10.62703354668001, −10.17482353698310, −9.530409203582620, −8.934683335913607, −8.657938769480787, −8.050988131952017, −7.449956623297515, −7.007372477721609, −6.358865898458310, −6.095588335040474, −5.196224317636048, −4.642732863900845, −4.191133094811268, −3.757531141446516, −2.801238236311056, −2.384865174160570, −1.716552816788143, −0.7402739988655993, 0, 0.7402739988655993, 1.716552816788143, 2.384865174160570, 2.801238236311056, 3.757531141446516, 4.191133094811268, 4.642732863900845, 5.196224317636048, 6.095588335040474, 6.358865898458310, 7.007372477721609, 7.449956623297515, 8.050988131952017, 8.657938769480787, 8.934683335913607, 9.530409203582620, 10.17482353698310, 10.62703354668001, 11.08683967984753, 11.59112170058487, 12.14187041615984, 12.46538042041295, 13.03044951056217, 13.63756382656552, 13.92209181688953

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