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Properties

Label 2-229320-1.1-c1-0-84
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·11-s + 13-s + 6·17-s − 8·19-s − 4·23-s + 25-s + 2·29-s − 4·31-s + 10·37-s + 6·41-s + 8·43-s + 8·47-s + 2·53-s + 4·55-s − 8·59-s + 2·61-s − 65-s − 4·67-s − 8·71-s + 10·73-s − 16·79-s + 12·83-s − 6·85-s − 18·89-s + 8·95-s + 2·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.20·11-s + 0.277·13-s + 1.45·17-s − 1.83·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s + 1.64·37-s + 0.937·41-s + 1.21·43-s + 1.16·47-s + 0.274·53-s + 0.539·55-s − 1.04·59-s + 0.256·61-s − 0.124·65-s − 0.488·67-s − 0.949·71-s + 1.17·73-s − 1.80·79-s + 1.31·83-s − 0.650·85-s − 1.90·89-s + 0.820·95-s + 0.203·97-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.04737125769077, −12.68119846785491, −12.31877499549146, −11.88793430478341, −11.13803285641914, −10.85156480111867, −10.46666219435928, −9.976784671430629, −9.475696098161572, −8.860954173738809, −8.372885315473411, −7.980174315595892, −7.457327346507716, −7.288725000653386, −6.233162887179279, −6.034907973343484, −5.571947440848133, −4.891208476802304, −4.266505385675167, −4.012004225650404, −3.294725490342954, −2.579958678719061, −2.346357278626911, −1.423162532770831, −0.7101227946545586, 0, 0.7101227946545586, 1.423162532770831, 2.346357278626911, 2.579958678719061, 3.294725490342954, 4.012004225650404, 4.266505385675167, 4.891208476802304, 5.571947440848133, 6.034907973343484, 6.233162887179279, 7.288725000653386, 7.457327346507716, 7.980174315595892, 8.372885315473411, 8.860954173738809, 9.475696098161572, 9.976784671430629, 10.46666219435928, 10.85156480111867, 11.13803285641914, 11.88793430478341, 12.31877499549146, 12.68119846785491, 13.04737125769077

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