Properties

Label 2-97020-1.1-c1-0-35
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 + 4·13-s + 2·17-s + 6·19-s + 6·23-s + 25-s − 4·29-s − 6·31-s + 2·37-s + 6·41-s + 4·43-s + 8·47-s + 6·53-s − 55-s + 12·59-s − 14·61-s − 4·65-s + 4·67-s + 12·73-s + 4·79-s − 2·85-s + 2·89-s − 6·95-s − 4·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s + 1.10·13-s + 0.485·17-s + 1.37·19-s + 1.25·23-s + 1/5·25-s − 0.742·29-s − 1.07·31-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 0.824·53-s − 0.134·55-s + 1.56·59-s − 1.79·61-s − 0.496·65-s + 0.488·67-s + 1.40·73-s + 0.450·79-s − 0.216·85-s + 0.211·89-s − 0.615·95-s − 0.406·97-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: \(0\)
Selberg data: \((2,\ 97020,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.365348219\)
\(L(\frac12)\) \(\approx\) \(3.365348219\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−13.82468466724492, −13.25114019398713, −12.84924263530129, −12.30543927440513, −11.79403993414708, −11.28379603325924, −10.92125893323740, −10.51154066632573, −9.706088719724374, −9.161432856085702, −9.022007122124445, −8.236275424122164, −7.705619722896286, −7.280758066108104, −6.821965663672538, −6.076715481484526, −5.522705707918152, −5.217360328081958, −4.311266194489695, −3.831628603001683, −3.351161997624943, −2.773907871919949, −1.911990327277122, −1.067487870861373, −0.7070149004286912, 0.7070149004286912, 1.067487870861373, 1.911990327277122, 2.773907871919949, 3.351161997624943, 3.831628603001683, 4.311266194489695, 5.217360328081958, 5.522705707918152, 6.076715481484526, 6.821965663672538, 7.280758066108104, 7.705619722896286, 8.236275424122164, 9.022007122124445, 9.161432856085702, 9.706088719724374, 10.51154066632573, 10.92125893323740, 11.28379603325924, 11.79403993414708, 12.30543927440513, 12.84924263530129, 13.25114019398713, 13.82468466724492

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