Properties

Label 2-229320-1.1-c1-0-20
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 − 3·11-s + 13-s + 7·17-s + 6·19-s − 3·23-s + 25-s + 6·29-s + 4·31-s − 5·37-s + 5·41-s − 2·43-s − 12·47-s − 2·53-s + 3·55-s − 59-s − 6·61-s − 65-s + 3·67-s − 11·73-s − 9·79-s − 4·83-s − 7·85-s + 11·89-s − 6·95-s − 17·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.904·11-s + 0.277·13-s + 1.69·17-s + 1.37·19-s − 0.625·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s − 0.821·37-s + 0.780·41-s − 0.304·43-s − 1.75·47-s − 0.274·53-s + 0.404·55-s − 0.130·59-s − 0.768·61-s − 0.124·65-s + 0.366·67-s − 1.28·73-s − 1.01·79-s − 0.439·83-s − 0.759·85-s + 1.16·89-s − 0.615·95-s − 1.72·97-s + 0.0995·101-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: \(0\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.916474076\)
\(L(\frac12)\) \(\approx\) \(1.916474076\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−12.94396636472117, −12.32090609874734, −12.02740695544745, −11.65812402373588, −11.13510990494250, −10.48465260063788, −10.14841280395490, −9.791750688785166, −9.271805579619443, −8.532629909903626, −8.149327820150109, −7.750884061585585, −7.427806296358685, −6.748049830392071, −6.225594777312696, −5.593140837758600, −5.260077965708441, −4.732748189330885, −4.130468826664703, −3.435512644340214, −3.035338200973401, −2.672390071085555, −1.608878350089196, −1.190149565772883, −0.4101861438555473, 0.4101861438555473, 1.190149565772883, 1.608878350089196, 2.672390071085555, 3.035338200973401, 3.435512644340214, 4.130468826664703, 4.732748189330885, 5.260077965708441, 5.593140837758600, 6.225594777312696, 6.748049830392071, 7.427806296358685, 7.750884061585585, 8.149327820150109, 8.532629909903626, 9.271805579619443, 9.791750688785166, 10.14841280395490, 10.48465260063788, 11.13510990494250, 11.65812402373588, 12.02740695544745, 12.32090609874734, 12.94396636472117

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