Properties

Label 2-229320-1.1-c1-0-46
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·11-s − 13-s + 4·19-s + 8·23-s + 25-s + 4·31-s + 8·37-s − 6·41-s − 2·43-s + 10·47-s − 6·53-s + 2·55-s + 12·59-s − 6·61-s + 65-s + 12·67-s + 2·71-s + 6·73-s − 4·79-s + 2·83-s − 14·89-s − 4·95-s + 10·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.603·11-s − 0.277·13-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.718·31-s + 1.31·37-s − 0.937·41-s − 0.304·43-s + 1.45·47-s − 0.824·53-s + 0.269·55-s + 1.56·59-s − 0.768·61-s + 0.124·65-s + 1.46·67-s + 0.237·71-s + 0.702·73-s − 0.450·79-s + 0.219·83-s − 1.48·89-s − 0.410·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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: \(0\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.661310514\)
\(L(\frac12)\) \(\approx\) \(2.661310514\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−12.96861253725933, −12.47223166523910, −11.98797472479651, −11.50624676804748, −11.14028379274715, −10.66095358683047, −10.15370172154796, −9.609975178236065, −9.290826316158962, −8.601344242816658, −8.256163372433114, −7.651185024642308, −7.340059919882448, −6.762351830887801, −6.359515069722232, −5.519448943871837, −5.220956256904687, −4.747460303487432, −4.131033056838089, −3.569240629783304, −2.835245469726321, −2.703372625332260, −1.787741993221743, −0.9717944806676009, −0.5450834393766349, 0.5450834393766349, 0.9717944806676009, 1.787741993221743, 2.703372625332260, 2.835245469726321, 3.569240629783304, 4.131033056838089, 4.747460303487432, 5.220956256904687, 5.519448943871837, 6.359515069722232, 6.762351830887801, 7.340059919882448, 7.651185024642308, 8.256163372433114, 8.601344242816658, 9.290826316158962, 9.609975178236065, 10.15370172154796, 10.66095358683047, 11.14028379274715, 11.50624676804748, 11.98797472479651, 12.47223166523910, 12.96861253725933

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