Properties

Label 2-97020-1.1-c1-0-46
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 − 6·13-s + 2·17-s + 2·19-s − 4·23-s + 25-s + 6·29-s − 2·37-s − 4·41-s + 4·43-s − 4·53-s − 55-s − 6·59-s − 2·61-s + 6·65-s + 14·67-s + 8·71-s + 10·73-s − 8·79-s + 12·83-s − 2·85-s − 14·89-s − 2·95-s − 10·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 1.66·13-s + 0.485·17-s + 0.458·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 0.328·37-s − 0.624·41-s + 0.609·43-s − 0.549·53-s − 0.134·55-s − 0.781·59-s − 0.256·61-s + 0.744·65-s + 1.71·67-s + 0.949·71-s + 1.17·73-s − 0.900·79-s + 1.31·83-s − 0.216·85-s − 1.48·89-s − 0.205·95-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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−14.13768862634196, −13.73134575087687, −12.89035516405834, −12.37606950924712, −12.17147437150125, −11.72943308556119, −11.11880165452944, −10.56855000818809, −9.982682401863955, −9.642679367801205, −9.202660569857204, −8.383057402076109, −8.015502068327803, −7.564056870899141, −6.934664527997220, −6.592688661439875, −5.836000934784390, −5.176872296380588, −4.839493085785642, −4.159126643291208, −3.611840474347876, −2.907734386033402, −2.403601174973463, −1.635202224535282, −0.7906359423752697, 0, 0.7906359423752697, 1.635202224535282, 2.403601174973463, 2.907734386033402, 3.611840474347876, 4.159126643291208, 4.839493085785642, 5.176872296380588, 5.836000934784390, 6.592688661439875, 6.934664527997220, 7.564056870899141, 8.015502068327803, 8.383057402076109, 9.202660569857204, 9.642679367801205, 9.982682401863955, 10.56855000818809, 11.11880165452944, 11.72943308556119, 12.17147437150125, 12.37606950924712, 12.89035516405834, 13.73134575087687, 14.13768862634196

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