Properties

Label 2-97020-1.1-c1-0-54
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 + 6·17-s + 4·19-s − 8·23-s + 25-s − 4·29-s + 4·31-s − 6·37-s − 6·41-s − 4·43-s + 4·47-s − 12·53-s + 55-s + 8·59-s + 4·61-s − 8·67-s + 12·71-s + 8·73-s − 8·79-s + 12·83-s − 6·85-s + 10·89-s − 4·95-s + 12·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s + 1.45·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.742·29-s + 0.718·31-s − 0.986·37-s − 0.937·41-s − 0.609·43-s + 0.583·47-s − 1.64·53-s + 0.134·55-s + 1.04·59-s + 0.512·61-s − 0.977·67-s + 1.42·71-s + 0.936·73-s − 0.900·79-s + 1.31·83-s − 0.650·85-s + 1.05·89-s − 0.410·95-s + 1.21·97-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 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−14.04378664077861, −13.65098567493974, −13.04037963925310, −12.43149457393382, −12.00091390296622, −11.76485572116846, −11.16461861659480, −10.44719436132473, −10.12640564953854, −9.653664120010598, −9.144994939579195, −8.284976772086320, −8.100172090931717, −7.560060144020053, −7.094579942450447, −6.363985793137320, −5.888770289993279, −5.182064179866905, −4.961407971383299, −4.004304643891872, −3.548556773613994, −3.146525558238922, −2.268134204777559, −1.624221619107739, −0.8411181709335724, 0, 0.8411181709335724, 1.624221619107739, 2.268134204777559, 3.146525558238922, 3.548556773613994, 4.004304643891872, 4.961407971383299, 5.182064179866905, 5.888770289993279, 6.363985793137320, 7.094579942450447, 7.560060144020053, 8.100172090931717, 8.284976772086320, 9.144994939579195, 9.653664120010598, 10.12640564953854, 10.44719436132473, 11.16461861659480, 11.76485572116846, 12.00091390296622, 12.43149457393382, 13.04037963925310, 13.65098567493974, 14.04378664077861

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