Properties

Label 2-97020-1.1-c1-0-44
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 − 7·13-s + 2·17-s − 2·23-s + 25-s − 2·29-s + 3·31-s + 12·37-s + 6·41-s + 43-s − 10·47-s − 55-s + 7·59-s + 7·65-s − 4·67-s − 9·71-s − 9·73-s − 6·79-s + 11·83-s − 2·85-s + 7·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 1.94·13-s + 0.485·17-s − 0.417·23-s + 1/5·25-s − 0.371·29-s + 0.538·31-s + 1.97·37-s + 0.937·41-s + 0.152·43-s − 1.45·47-s − 0.134·55-s + 0.911·59-s + 0.868·65-s − 0.488·67-s − 1.06·71-s − 1.05·73-s − 0.675·79-s + 1.20·83-s − 0.216·85-s + 0.741·89-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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 \( 1 + 7 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.32158069105418, −13.48683372104993, −12.82984902310394, −12.72135418678723, −11.90143637858153, −11.69298936599601, −11.30415955364523, −10.37388739447666, −10.17039138059311, −9.521115948369796, −9.232742607872331, −8.501949915791179, −7.819541999568643, −7.589441818127378, −7.140471663620019, −6.338202959661422, −5.999519809893431, −5.155177418451127, −4.770115817545825, −4.216208473456613, −3.640080765340650, −2.780482975180050, −2.507161973040800, −1.627004668483553, −0.7792931789645702, 0, 0.7792931789645702, 1.627004668483553, 2.507161973040800, 2.780482975180050, 3.640080765340650, 4.216208473456613, 4.770115817545825, 5.155177418451127, 5.999519809893431, 6.338202959661422, 7.140471663620019, 7.589441818127378, 7.819541999568643, 8.501949915791179, 9.232742607872331, 9.521115948369796, 10.17039138059311, 10.37388739447666, 11.30415955364523, 11.69298936599601, 11.90143637858153, 12.72135418678723, 12.82984902310394, 13.48683372104993, 14.32158069105418

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