Properties

Label 2-97020-1.1-c1-0-34
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 − 6·17-s − 8·19-s + 6·23-s + 25-s − 6·29-s − 2·31-s + 2·37-s + 8·43-s − 12·47-s − 6·53-s − 55-s + 6·59-s − 8·61-s + 2·65-s + 2·67-s + 10·73-s + 8·79-s + 12·83-s + 6·85-s + 6·89-s + 8·95-s − 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 0.554·13-s − 1.45·17-s − 1.83·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s − 0.359·31-s + 0.328·37-s + 1.21·43-s − 1.75·47-s − 0.824·53-s − 0.134·55-s + 0.781·59-s − 1.02·61-s + 0.248·65-s + 0.244·67-s + 1.17·73-s + 0.900·79-s + 1.31·83-s + 0.650·85-s + 0.635·89-s + 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.10754296544320, −13.38132622088177, −12.95856864554648, −12.69827961871305, −12.16205659861651, −11.38262468030053, −11.04967875178996, −10.86265826958120, −10.09841508638417, −9.473166686915669, −8.941874803770184, −8.738643902403676, −7.974680881825163, −7.515293740390228, −6.943295659506878, −6.386364064421590, −6.131969099774426, −5.027216041887357, −4.843477110860442, −4.146401534757653, −3.691966904464821, −2.944763055231988, −2.191909373671666, −1.836817580795059, −0.7049085794792177, 0, 0.7049085794792177, 1.836817580795059, 2.191909373671666, 2.944763055231988, 3.691966904464821, 4.146401534757653, 4.843477110860442, 5.027216041887357, 6.131969099774426, 6.386364064421590, 6.943295659506878, 7.515293740390228, 7.974680881825163, 8.738643902403676, 8.941874803770184, 9.473166686915669, 10.09841508638417, 10.86265826958120, 11.04967875178996, 11.38262468030053, 12.16205659861651, 12.69827961871305, 12.95856864554648, 13.38132622088177, 14.10754296544320

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