Properties

Label 2-97020-1.1-c1-0-47
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 − 5·13-s + 6·17-s + 4·19-s − 6·23-s + 25-s − 6·29-s − 5·31-s − 4·37-s + 11·43-s − 6·53-s − 55-s − 3·59-s + 10·61-s + 5·65-s + 8·67-s − 3·71-s + 13·73-s + 8·79-s + 3·83-s − 6·85-s − 15·89-s − 4·95-s + 10·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 1.38·13-s + 1.45·17-s + 0.917·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s − 0.898·31-s − 0.657·37-s + 1.67·43-s − 0.824·53-s − 0.134·55-s − 0.390·59-s + 1.28·61-s + 0.620·65-s + 0.977·67-s − 0.356·71-s + 1.52·73-s + 0.900·79-s + 0.329·83-s − 0.650·85-s − 1.58·89-s − 0.410·95-s + 1.01·97-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 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.02922502254205, −13.78786657078580, −12.79879701198443, −12.50143296601854, −12.13010715760716, −11.70871668218752, −11.08654799708409, −10.66716564275890, −9.836445308701723, −9.685982764736633, −9.261145825845623, −8.439307185201111, −7.880485428574699, −7.520021198424294, −7.171666969335998, −6.471834376368147, −5.680669619324619, −5.387398082435319, −4.849879712770855, −3.949186638432371, −3.729124072621644, −2.994983344273491, −2.311486772120950, −1.649225904675634, −0.8082685282704251, 0, 0.8082685282704251, 1.649225904675634, 2.311486772120950, 2.994983344273491, 3.729124072621644, 3.949186638432371, 4.849879712770855, 5.387398082435319, 5.680669619324619, 6.471834376368147, 7.171666969335998, 7.520021198424294, 7.880485428574699, 8.439307185201111, 9.261145825845623, 9.685982764736633, 9.836445308701723, 10.66716564275890, 11.08654799708409, 11.70871668218752, 12.13010715760716, 12.50143296601854, 12.79879701198443, 13.78786657078580, 14.02922502254205

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