Properties

Label 2-229320-1.1-c1-0-121
Degree $2$
Conductor $229320$
Sign $1$
Analytic cond. $1831.12$
Root an. cond. $42.7916$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·11-s + 13-s + 2·19-s − 8·23-s + 25-s + 2·29-s − 2·31-s − 8·37-s − 6·41-s − 10·43-s + 2·47-s − 2·53-s + 4·55-s + 8·59-s − 65-s + 4·67-s − 10·73-s − 16·79-s − 14·83-s − 2·89-s − 2·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.20·11-s + 0.277·13-s + 0.458·19-s − 1.66·23-s + 1/5·25-s + 0.371·29-s − 0.359·31-s − 1.31·37-s − 0.937·41-s − 1.52·43-s + 0.291·47-s − 0.274·53-s + 0.539·55-s + 1.04·59-s − 0.124·65-s + 0.488·67-s − 1.17·73-s − 1.80·79-s − 1.53·83-s − 0.211·89-s − 0.205·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(229320\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1831.12\)
Root analytic conductor: \(42.7916\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{229320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ 1)\)

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 \( 1 \)
13 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−13.39566608565912, −12.86972989659417, −12.54458668792787, −11.90943315283497, −11.54772577932032, −11.25058049518285, −10.37164060314052, −10.13692961862515, −10.01021211351532, −9.057320172897060, −8.526144664019689, −8.358391393447138, −7.631395735765082, −7.425203285367212, −6.729669596472549, −6.287062913956636, −5.535726459573260, −5.316808649224243, −4.687331762196869, −4.105517621602797, −3.528688461062444, −3.106893176472216, −2.416068601358310, −1.827690618351708, −1.186034292375104, 0, 0, 1.186034292375104, 1.827690618351708, 2.416068601358310, 3.106893176472216, 3.528688461062444, 4.105517621602797, 4.687331762196869, 5.316808649224243, 5.535726459573260, 6.287062913956636, 6.729669596472549, 7.425203285367212, 7.631395735765082, 8.358391393447138, 8.526144664019689, 9.057320172897060, 10.01021211351532, 10.13692961862515, 10.37164060314052, 11.25058049518285, 11.54772577932032, 11.90943315283497, 12.54458668792787, 12.86972989659417, 13.39566608565912

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