Properties

Label 2-97020-1.1-c1-0-33
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 about

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 11-s − 6·13-s + 2·17-s − 8·19-s + 8·23-s + 25-s − 6·29-s − 2·37-s − 6·41-s + 8·43-s + 8·47-s − 2·53-s + 55-s − 6·59-s + 8·61-s + 6·65-s + 10·67-s − 8·71-s − 14·73-s + 10·79-s − 18·83-s − 2·85-s + 14·89-s + 8·95-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s − 1.66·13-s + 0.485·17-s − 1.83·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s − 0.328·37-s − 0.937·41-s + 1.21·43-s + 1.16·47-s − 0.274·53-s + 0.134·55-s − 0.781·59-s + 1.02·61-s + 0.744·65-s + 1.22·67-s − 0.949·71-s − 1.63·73-s + 1.12·79-s − 1.97·83-s − 0.216·85-s + 1.48·89-s + 0.820·95-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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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

−14.25313219807174, −13.33191098246454, −13.00942891536756, −12.57551447291631, −12.15222202383727, −11.62981229619056, −11.01349796147080, −10.60221786421205, −10.20177766688694, −9.516714775162448, −9.056405281689939, −8.595152644739903, −7.966329619933107, −7.409176697943531, −7.099571240550861, −6.566508456378802, −5.751517295948228, −5.313034968764035, −4.670981029289811, −4.302785238961429, −3.565375306880712, −2.890942310026862, −2.367199763978805, −1.738470880027815, −0.7004744785182929, 0, 0.7004744785182929, 1.738470880027815, 2.367199763978805, 2.890942310026862, 3.565375306880712, 4.302785238961429, 4.670981029289811, 5.313034968764035, 5.751517295948228, 6.566508456378802, 7.099571240550861, 7.409176697943531, 7.966329619933107, 8.595152644739903, 9.056405281689939, 9.516714775162448, 10.20177766688694, 10.60221786421205, 11.01349796147080, 11.62981229619056, 12.15222202383727, 12.57551447291631, 13.00942891536756, 13.33191098246454, 14.25313219807174

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