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Properties

Label 2-97020-1.1-c1-0-9
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 11-s + 6·13-s + 2·17-s − 6·23-s + 25-s − 6·29-s + 2·31-s + 10·37-s − 8·41-s − 8·43-s + 4·47-s − 6·53-s + 55-s − 6·59-s + 8·61-s − 6·65-s + 14·67-s + 8·71-s + 2·73-s − 16·79-s + 12·83-s − 2·85-s − 18·89-s − 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s + 1.66·13-s + 0.485·17-s − 1.25·23-s + 1/5·25-s − 1.11·29-s + 0.359·31-s + 1.64·37-s − 1.24·41-s − 1.21·43-s + 0.583·47-s − 0.824·53-s + 0.134·55-s − 0.781·59-s + 1.02·61-s − 0.744·65-s + 1.71·67-s + 0.949·71-s + 0.234·73-s − 1.80·79-s + 1.31·83-s − 0.216·85-s − 1.90·89-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 97020,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.823084011\)
\(L(\frac12)\) \(\approx\) \(1.823084011\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−13.76722015054590, −13.33505940273561, −12.78704830450731, −12.40750087612431, −11.67421481302437, −11.33884312092875, −10.99574877045893, −10.32704378557529, −9.821817428270046, −9.421230426605246, −8.600201715891644, −8.274796443742915, −7.907941410628739, −7.317298625139300, −6.529367715356952, −6.275658592605086, −5.542145413429289, −5.180789718061179, −4.239199983843680, −3.908541418238375, −3.376595009959258, −2.719795087755545, −1.868913731888509, −1.289626914077572, −0.4441359903266406, 0.4441359903266406, 1.289626914077572, 1.868913731888509, 2.719795087755545, 3.376595009959258, 3.908541418238375, 4.239199983843680, 5.180789718061179, 5.542145413429289, 6.275658592605086, 6.529367715356952, 7.317298625139300, 7.907941410628739, 8.274796443742915, 8.600201715891644, 9.421230426605246, 9.821817428270046, 10.32704378557529, 10.99574877045893, 11.33884312092875, 11.67421481302437, 12.40750087612431, 12.78704830450731, 13.33505940273561, 13.76722015054590

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