Properties

Label 2-97020-1.1-c1-0-14
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 + 5·13-s + 5·19-s + 6·23-s + 25-s − 31-s − 7·37-s − 43-s − 12·47-s + 6·53-s + 55-s − 6·59-s − 10·61-s − 5·65-s − 13·67-s − 12·71-s + 5·73-s − 79-s − 5·95-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 6·115-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s + 1.38·13-s + 1.14·19-s + 1.25·23-s + 1/5·25-s − 0.179·31-s − 1.15·37-s − 0.152·43-s − 1.75·47-s + 0.824·53-s + 0.134·55-s − 0.781·59-s − 1.28·61-s − 0.620·65-s − 1.58·67-s − 1.42·71-s + 0.585·73-s − 0.112·79-s − 0.512·95-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 − 0.559·115-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

\(L(1)\) \(\approx\) \(2.154195293\)
\(L(\frac12)\) \(\approx\) \(2.154195293\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.77993880841411, −13.21510095248225, −13.00209712718312, −12.18983959066586, −11.87026067922829, −11.24134861249639, −10.96629613463605, −10.38179965267883, −9.914012473729736, −9.121124666194077, −8.863079610410703, −8.349144618407812, −7.685420602842065, −7.321974502950329, −6.736671008495697, −6.123615258400583, −5.630785804223135, −4.956314704208659, −4.570872373826679, −3.729154308773699, −3.224292719240764, −2.936977611690190, −1.775190146058172, −1.303838993376178, −0.4894409854683823, 0.4894409854683823, 1.303838993376178, 1.775190146058172, 2.936977611690190, 3.224292719240764, 3.729154308773699, 4.570872373826679, 4.956314704208659, 5.630785804223135, 6.123615258400583, 6.736671008495697, 7.321974502950329, 7.685420602842065, 8.349144618407812, 8.863079610410703, 9.121124666194077, 9.914012473729736, 10.38179965267883, 10.96629613463605, 11.24134861249639, 11.87026067922829, 12.18983959066586, 13.00209712718312, 13.21510095248225, 13.77993880841411

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