Properties

Label 2-97020-1.1-c1-0-0
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 − 2·13-s − 2·19-s + 25-s − 8·31-s + 2·37-s + 2·43-s − 6·53-s + 55-s − 12·59-s − 2·61-s + 2·65-s − 4·67-s − 2·73-s − 10·79-s − 12·83-s − 6·89-s + 2·95-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s − 0.554·13-s − 0.458·19-s + 1/5·25-s − 1.43·31-s + 0.328·37-s + 0.304·43-s − 0.824·53-s + 0.134·55-s − 1.56·59-s − 0.256·61-s + 0.248·65-s − 0.488·67-s − 0.234·73-s − 1.12·79-s − 1.31·83-s − 0.635·89-s + 0.205·95-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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: \(0\)
Selberg data: \((2,\ 97020,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5590977488\)
\(L(\frac12)\) \(\approx\) \(0.5590977488\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.86715974627571, −13.19723793090289, −12.67451962876442, −12.49583487375313, −11.79338306382370, −11.34768313764213, −10.79894074055829, −10.48569497208347, −9.759278140037112, −9.347508219997532, −8.814216866169924, −8.236603447151385, −7.746807352637289, −7.258474016182098, −6.837081125782210, −6.066042594665715, −5.646127737070737, −4.979166150744997, −4.426580929912465, −3.972614806985850, −3.178183820455965, −2.749754139891258, −1.944420559788206, −1.323584017366382, −0.2342953477821970, 0.2342953477821970, 1.323584017366382, 1.944420559788206, 2.749754139891258, 3.178183820455965, 3.972614806985850, 4.426580929912465, 4.979166150744997, 5.646127737070737, 6.066042594665715, 6.837081125782210, 7.258474016182098, 7.746807352637289, 8.236603447151385, 8.814216866169924, 9.347508219997532, 9.759278140037112, 10.48569497208347, 10.79894074055829, 11.34768313764213, 11.79338306382370, 12.49583487375313, 12.67451962876442, 13.19723793090289, 13.86715974627571

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